CN105182801B - A kind of Stewart platform active vibration isolation PD control methods based on extended state observer - Google Patents
A kind of Stewart platform active vibration isolation PD control methods based on extended state observer Download PDFInfo
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Abstract
A kind of Stewart platform active vibration isolation PD control methods based on extended state observer, the present invention relates to PD control method.The present invention is to solve the formulation of control strategy is relatively simple, control accuracy have much room for improvement, do not account for system uncertain flexible appendage influence, do not account for the structural nonlinear of platform and the problem of the design process of control algolithm has arbitrariness and a kind of Stewart platform active vibration isolation PD control methods based on extended state observer for proposing.This method is by one, sets up the kinetic models of Stewart platforms；2nd, the kinetic model of six executing agencies of Stewart platforms is set up；3rd, the state space of Stewart platforms is obtained；4th, it is convergent observation error to the observation error of system mode to determine observer；5th, the step such as the PD control device of design based on expansion observer is realized.The present invention is applied to PD control method field.
Description
Technical field
It is more particularly to a kind of based on expansion state observation the present invention relates to Stewart platform active vibration isolation PD control methods
The Stewart platform active vibration isolation PD control methods of device.
Background technology
The space tasks that current spacecraft faces are more and more varied, it is necessary to carry substantial amounts of measurement or communication apparatus,
Therefore cause spacecraft self structure increasingly complicated, and low frequency modal is intensive, while these sensitive loads loaded often have
Very high pointing accuracy and stability demand.It has been difficult to complete more next if carrying out gesture stability just for spacecraft body
Higher mission requirements comes to ensureing the stability of sensitive load, it is necessary to seek other schemes, and carries out multiple degrees of freedom to it
Vibration isolation is the focus of recent research, and Stewart platforms are due to its special architectural characteristic in numerous schemes so that it is fitted very much
Close and 6DOF isolating technique and microoperation are carried out to sensitive load, domestic and international many scholars have made intensive studies to this.
Document Su Y X, Duan B Y, Zheng C H.Nonlinear PID control of a six-DOF
parallel manipulator[J].IEE Proceedings-Control Theory and Applications,2004,
151(1):95-102. proposes a kind of nonlinear PID controller method, is tracked with the high-precision attitude for realizing Stewart platforms.
The kinematics model of six degree of freedom Stewart platform is set up, and nonlinear PID controller method is used based on the model, it is most laggard
Experimental verification is gone.But process is set up as can be seen that the program is only modeled to joint space, not from kinematical equation
Have and Dynamic Modeling analysis is carried out to operating platform kinetic characteristic, due to using traditional PID control as core control program,
Cause control effect unsatisfactory.
Document Zhang Y, Zhang J.The imaging stability enhancement of optical
payload using multiple vibration isolation platforms[J].Journal of Vibration&
Control, 2013. have highlighted the spacecraft modeling scheme containing multiple Steward vibration-isolating platforms, and to having carried out frequency domain
And stability analysis.But the vibration isolating method used in text belongs to the category of passive vibration isolation, it is only capable of isolating the vibration of specific frequency, no
The vibration of Whole frequency band can be isolated, vibration isolating effect is limited, while abbreviation largely has been carried out in modeling process,
The influence for not modeling uncertain and flexible appendage of system is not accounted for.
Document Yang T, Ma J, Hou Z G, et al.Robust back stepping control of active
vibration isolation using a stewart platform[C]//IEEE International
Conference on Robotics and Automation.IEEE,2009:1788-1793. utilizes Newton-Euler side
Method establishes the platform model as actuator using voice coil motor, and carries out decoupling processing to platform, and its is equivalent into 6 lists
Enter the passage singly gone out and devise the nonlinear robust control algorithm based on Lyapunov stability, to overcome the uncertain of system
Property.But the structural nonlinear that the geometrical non-linearity of platform does not consider platform is only accounted in text, also not to uncertainty
Source analysed in depth, and the design process of control algolithm has arbitrariness.
The content of the invention
The invention aims to solve prior art not carry out detailed modeling analysis, control strategy to operating platform
Formulation it is relatively simple, control accuracy has much room for improvement, control effect is limited, do not account for the uncertain flexible appendage of system
One for influenceing, not accounting for the problem of structural nonlinear of platform and the design process of control algolithm have arbitrariness and propose
Plant Stewart platform active vibration isolations PD (proportion differentiation) controlling party based on extended state observer
Method.
Above-mentioned goal of the invention is achieved through the following technical solutions：
Step 1: setting plane on Stewart platforms around the X-axis of Stewart platforms with angleRotated, Stewart is put down
Plane is rotated around the Y-axis of Stewart platforms with angle, θ on platform, Z axis of the plane around Stewart platforms on Stewart platforms
Rotated with angle ψ, shown in the kinetic model such as formula (1) that Stewart platforms are set up by Newton-Euler methods：
Wherein,For the broad sense position vector on Stewart platforms, x, y and z are respectively
Displacement of the plane on X, Y and Z axis on Stewart platforms；M∈R6×6For the inertial matrix of Stewart platforms；R is real number；B∈
R6×6For the damping matrix of Stewart platforms；K∈R6×6For the stiffness matrix of Stewart platforms；C∈R6For Stewart platforms institute
By centripetal force；For Stewart platforms coriolis acceleration vector and be χ first derivatives；Add for Stewart platforms Ke Shi
Velocity vector and for χ second dervatives；Δs∈R6For model uncertainty, τ ∈ R6The broad sense produced for the executing agency of 6 poles
Driving force；ws∈R6Vector is disturbed caused by external vibration；
Step 2: according to the voltage equation of voice coil loudspeaker voice coil motor, setting up six executing agencies of Stewart platforms
Shown in kinetic model such as formula (2)：
Wherein, L is voice coil motor inductance coefficent matrix；R is the resistor matrix of voice coil motor, KeFor the anti-electricity of voice coil motor
Kinetic potential matrix；Δm∈R6For the uncertainty of voice coil motor；U is the PD control device based on extended state observer；wm∈R6For
Interference vector caused by external vibration in motor；J is Jacobian matrix；imFor the current strength of voice coil motor coil；For im's
First derivative；
Step 3: according to the kinetic model of Stewart platforms and the dynamics of six executing agencies of Stewart platforms
Model calculates the state space for obtaining Stewart platforms；
Step 4: by the state space of Stewart platforms, design Stewart platforms extended state observer calculates expansion
State observerDetermine that extended state observer is missed to the observation error of system mode for convergent observation
Difference；
Step 5: according to the observed quantity of expansion observerThe PD control device based on extended state observer is designed,
Suppress extraneous interference disturbance to carry out the vibration isolation control of Stewart platforms using PD control device；
Wherein, kpFor the proportionality coefficient of PD control device；kdFor the differential coefficient of PD control device；x1=χ；For x2Observation
State,For x4Observer state；PD control implement body based on extended state observer is：
Invention effect
The present invention considers 6DOF vibration isolation problem, devises the master of the Stewart platforms based on cube configuration
Passive vibration isolation algorithm, initially sets up the kinematics and dynamics modeling of Stewart platforms using voice coil motor as executing agency simultaneously
Model is rationally converted, by the equivalent subsystem into 6 single-input single-outputs of platform, it is contemplated that sensor in practical application
The Cost Problems of quantity and equipment, devise the PD (proportion differentiation) based on extended state observer
Vibration isolation controller, and give parameter design rule when considering extended state observer convergence, designed control
Algorithm processed drastically increases the vibration isolating effect of platform while the number of sensor needed for effectively reducing compared with passive vibration isolation
Amount.
Compared with above-mentioned algorithm, the present invention has advantages below：
1. many methods all do not account for executing agency, and the present invention establishes the dynamics of voice coil motor actuator in detail
Equation；
2. most of control method is all based on what joint space was designed, and control accuracy is limited, and the present invention is based on
Working space is controlled, and control accuracy is higher；
3. the present invention considers the immeasurability of the platform generalized velocity of the control algolithm based on workplace design, using expansion
Open state observer to observe generalized velocity information, reduce the complexity and cost of operating platform in practical application.
From Fig. 6~8, extended state observer can track broad sense position and the speed of system within the limited time
Signal, and the state is expanded it is estimated that external disturbance.In the case where only considering passive vibration isolation, cube of the invention
The Stewart platforms of configuration have good vibration isolating effect to the sinusoidal interference and random noise disturbance in broadband.Add base
In after the PD active vibration isolations of extended state observer, vibration isolating effect all improves significantly, and steady-state error has an order of magnitude
Lifting, the especially sinusoidal interference and random noise disturbance to medium-high frequency.
The kinetic model for setting up Stewart platforms and six executing agencies of Stewart platforms by the present invention
Kinetic model process can see, it is contemplated that the structural nonlinear of platform, the PD of the invention based on extended state observer
In the condition of convergence that controller design is provided, provide the requirement of specific value, it is to avoid the arbitrariness of algorithm.
Brief description of the drawings
Fig. 1 is the vector table diagram for the Stewart platforms that embodiment one is proposed；Wherein, upper mounting plate and 6 poles
Tie point be end device Ai, i=1,2,3, the tie point of lower platform and 6 poles is Bi, i=1,2,3；{ B } is to be fixed on down
The inertial coordinate system of platform, its origin is overlapped with the barycenter of lower platform, and { P } is the reference frame of moving platform；rbaseIt is
The origin of { B } is to pedestal tie point BiRadial distance, rendIt is the origin of { P } to pedestal tie point AiRadial distance；
Fig. 2 is a kind of Stewart platform active vibration isolations based on extended state observer that embodiment one is proposed
PD control method；
Fig. 3 (a) is that the interference that embodiment is proposed is that amplitude is 1, translation (open loop) x when frequency is 10Hz sinusoidal signal
Direction displacement；
Fig. 3 (b) is that the interference that embodiment is proposed is that amplitude is 1, the rotation (open loop) when frequency is 10Hz sinusoidal signal
Direction displacement；
Fig. 3 (c) is that the interference that embodiment is proposed is that amplitude is 1, translation (open loop) y when frequency is 10Hz sinusoidal signal
Direction displacement；
Fig. 3 (d) is that the interference that embodiment is proposed is that amplitude is 1, rotation (open loop) θ when frequency is 10Hz sinusoidal signal
Direction displacement；
Fig. 3 (e) is that the interference that embodiment is proposed is that amplitude is 1, translation (open loop) z when frequency is 10Hz sinusoidal signal
Direction displacement；
Fig. 3 (f) is that the interference that embodiment is proposed is that amplitude is 1, rotation (open loop) ψ when frequency is 10Hz sinusoidal signal
Direction displacement；
Fig. 4 (a) is that the interference that embodiment is proposed is that amplitude is 1, translation (closed loop) x when frequency is 10Hz sinusoidal signal
Direction displacement；
Fig. 4 (b) is that the interference that embodiment is proposed is that amplitude is 1, the rotation (closed loop) when frequency is 10Hz sinusoidal signalDirection displacement；
Fig. 4 (c) is that the interference that embodiment is proposed is that amplitude is 1, translation (closed loop) y when frequency is 10Hz sinusoidal signal
Direction displacement；
Fig. 4 (d) is that the interference that embodiment is proposed is that amplitude is 1, rotation (closed loop) θ when frequency is 10Hz sinusoidal signal
Direction displacement；
Fig. 4 (e) is that the interference that embodiment is proposed is that amplitude is 1, translation (closed loop) z when frequency is 10Hz sinusoidal signal
Direction displacement；
Fig. 4 (f) is that the interference that embodiment is proposed is that amplitude is 1, rotation (closed loop) ψ when frequency is 10Hz sinusoidal signal
Direction displacement；
Fig. 5 (a) is the l that embodiment is proposed1Pole length change curve schematic diagram；
Fig. 5 (b) is the l that embodiment is proposed4Pole length change curve schematic diagram；
Fig. 5 (c) is the l that embodiment is proposed2Pole length change curve schematic diagram；
Fig. 5 (d) is the l that embodiment is proposed5Pole length change curve schematic diagram；
Fig. 5 (e) is the l that embodiment is proposed3Pole length change curve schematic diagram；
Fig. 5 (f) is the l that embodiment is proposed6Pole length change curve schematic diagram；
The estimation that Fig. 6 (a) is the generalized displacement x ' of the real displacement curve x that embodiment is proposed and extended state observer is bent
Line；
Fig. 6 (b) is the real angle curve that embodiment is proposedWith the broadest scope of extended state observerEstimation it is bent
Line；
The estimation that Fig. 6 (c) is the generalized displacement y ' of the real displacement curve y that embodiment is proposed and extended state observer is bent
Line；
The estimation that Fig. 6 (d) is the broadest scope θ ' of the real angle curve θ that embodiment is proposed and extended state observer is bent
Line；
The estimation that Fig. 6 (e) is the generalized displacement z ' of the real displacement curve z that embodiment is proposed and extended state observer is bent
Line；
The estimation that Fig. 6 (f) is the broadest scope ψ ' of the real angle curve ψ that embodiment is proposed and extended state observer is bent
Line；
Fig. 7 (a) is the generalized velocity dx ' of the true velocity curve dx that embodiment is proposed and extended state observer estimation
Curve；
The true angular velocity curve that Fig. 7 (b) embodiments are proposedWith the broad sense angular speed of extended state observerEstimate
Index contour；
Fig. 7 (c) is the generalized velocity dy ' of the true velocity curve dy that embodiment is proposed and extended state observer estimation
Curve；
Fig. 7 (d) is the broad sense angular speed d θ's ' of the true angular velocity curve d θ that embodiment is proposed and extended state observer
Estimation curve；
Fig. 7 (e) is the generalized velocity dz ' of the true velocity curve dz that embodiment is proposed and extended state observer estimation
Curve；
Fig. 7 (f) is the broad sense angular speed d ψ ' of the true angular velocity curve d ψ that embodiment is proposed and extended state observer
Estimation curve；
Fig. 8 (a) is the true interference curve d that embodiment is proposedxD ' is disturbed with the broad sense of extended state observerxEstimation
Curve；
The true interference curve that Fig. 8 (b) embodiments are proposedDisturbed with the broad sense of extended state observerEstimation it is bent
Line；
Fig. 8 (c) is the true interference curve d that embodiment is proposedyD ' is disturbed with the broad sense of extended state observeryEstimation
Curve；
Fig. 8 (d) is the true interference curve d that embodiment is proposedθD ' is disturbed with the broad sense of extended state observerθEstimation
Curve；
Fig. 8 (e) is the true interference curve d that embodiment is proposedzD ' is disturbed with the broad sense of extended state observerzEstimation
Curve；
Fig. 8 (f) is the true interference curve d that embodiment is proposedψD ' is disturbed with the broad sense of extended state observerψEstimate
Index contour；
Fig. 9 (a) is that the interference that embodiment is proposed is that amplitude is 1, translation (open loop) x when frequency is 50Hz sinusoidal signal
Direction displacement；
Fig. 9 (b) is that the interference that embodiment is proposed is that amplitude is 1, the rotation (open loop) when frequency is 50Hz sinusoidal signal
Direction displacement；
Fig. 9 (c) is that the interference that embodiment is proposed is that amplitude is 1, translation (open loop) y when frequency is 50Hz sinusoidal signal
Direction displacement；
Fig. 9 (d) is that the interference that embodiment is proposed is that amplitude is 1, rotation (open loop) θ when frequency is 50Hz sinusoidal signal
Direction displacement；
Fig. 9 (e) is that the interference that embodiment is proposed is that amplitude is 1, translation (open loop) z when frequency is 50Hz sinusoidal signal
Direction displacement；
Fig. 9 (f) is that the interference that embodiment is proposed is that amplitude is 1, rotation (open loop) ψ when frequency is 50Hz sinusoidal signal
Direction displacement；
Figure 10 (a) is that the interference that embodiment is proposed is that amplitude is 1, the translation (closed loop) when frequency is 50Hz sinusoidal signal
The displacement of x directions；
Figure 10 (b) is that the interference that embodiment is proposed is that amplitude is 1, the rotation (closed loop) when frequency is 50Hz sinusoidal signalDirection displacement；
Figure 10 (c) is that the interference that embodiment is proposed is that amplitude is 1, the translation (closed loop) when frequency is 50Hz sinusoidal signal
The displacement of y directions；
Figure 10 (d) is that the interference that embodiment is proposed is that amplitude is 1, the rotation (closed loop) when frequency is 50Hz sinusoidal signal
The displacement of θ directions；
Figure 10 (e) is that the interference that embodiment is proposed is that amplitude is 1, the translation (closed loop) when frequency is 50Hz sinusoidal signal
The displacement of z directions；
Figure 10 (f) is that the interference that embodiment is proposed is that amplitude is 1, the rotation (closed loop) when frequency is 50Hz sinusoidal signal
The displacement of ψ directions；
Figure 11 (a) is that the interference that embodiment is proposed is that amplitude is 1, and translation when frequency is 100Hz sinusoidal signal (is opened
Ring) displacement of x directions；
Figure 11 (b) is that the interference that embodiment is proposed is that amplitude is 1, and rotation when frequency is 100Hz sinusoidal signal (is opened
Ring)Direction displacement；
Figure 11 (c) is that the interference that embodiment is proposed is that amplitude is 1, and translation when frequency is 100Hz sinusoidal signal (is opened
Ring) displacement of y directions；
Figure 11 (d) is that the interference that embodiment is proposed is that amplitude is 1, and rotation when frequency is 100Hz sinusoidal signal (is opened
Ring) displacement of θ directions；
Figure 11 (e) is that the interference that embodiment is proposed is that amplitude is 1, and translation when frequency is 100Hz sinusoidal signal (is opened
Ring) displacement of z directions；
Figure 11 (f) is that the interference that embodiment is proposed is that amplitude is 1, rotation displacement when frequency is 100Hz sinusoidal signal
(open loop) ψ direction displacements；
Figure 12 (a) is that the interference that embodiment is proposed is that amplitude is 1, and translation when frequency is 100Hz sinusoidal signal (is closed
Ring) displacement of x directions；
Figure 12 (b) is that the interference that embodiment is proposed is that amplitude is 1, and rotation when frequency is 100Hz sinusoidal signal (is closed
Ring)Direction displacement；
Figure 12 (c) is that the interference that embodiment is proposed is that amplitude is 1, and translation when frequency is 100Hz sinusoidal signal (is closed
Ring) displacement of y directions；
Figure 12 (d) is that the interference that embodiment is proposed is that amplitude is 1, and rotation when frequency is 100Hz sinusoidal signal (is closed
Ring) displacement of θ directions；
Figure 12 (e) is that the interference that embodiment is proposed is that amplitude is 1, and translation when frequency is 100Hz sinusoidal signal (is closed
Ring) displacement of z directions；
Figure 12 (f) is that the interference that embodiment is proposed is that amplitude is 1, rotation displacement when frequency is 100Hz sinusoidal signal
(closed loop) ψ direction displacements；
Figure 13 (a) is that the interference that embodiment is proposed is that (average is 0 to random noise, and variance is translation (open loop) x when 25)
Direction displacement；
Figure 13 (b) is that the interference that embodiment is proposed is that (average is 0 to random noise, and variance is that rotation displacement when 25) (is opened
Ring)Direction displacement；
Figure 13 (c) is that the interference that embodiment is proposed is that (average is 0 to random noise, and variance is translation (open loop) y when 25)
Direction displacement；
Figure 13 (d) is that the interference that embodiment is proposed is that (average is 0 to random noise, and variance is the rotation (open loop) when 25)
The displacement of θ directions；
Figure 13 (e) is that the interference that embodiment is proposed is that (average is 0 to random noise, and variance is (open loop) of translation when 25)
The displacement of z directions；
Figure 13 (f) is that the interference that embodiment is proposed is that (average is 0 to random noise, and variance is that rotation displacement when 25) (is opened
Ring) the displacement of ψ directions；
Figure 14 (a) is that the interference that embodiment is proposed is that (average is 0 to random noise, and variance is translation (closed loop) x when 25)
Direction displacement；
Figure 14 (b) is that the interference that embodiment is proposed is that (average is 0 to random noise, and variance is that rotation displacement when 25) (is closed
Ring)Direction displacement；
Figure 14 (c) is that the interference that embodiment is proposed is that (average is 0 to random noise, and variance is translation (closed loop) y when 25)
Direction displacement；
Figure 14 (d) is that the interference that embodiment is proposed is that (average is 0 to random noise, and variance is the rotation (closed loop) when 25)
The displacement of θ directions；
Figure 14 (e) is that the interference that embodiment is proposed is that (average is 0 to random noise, and variance is (closed loop) of translation when 25)
The displacement of z directions；
Figure 14 (f) is that the interference that embodiment is proposed is that (average is 0 to random noise, and variance is that rotation displacement when 25) (is closed
Ring) the displacement of ψ directions.
Embodiment
Embodiment one：With reference to a kind of Stewart platforms based on extended state observer of Fig. 2 present embodiments
Active vibration isolation PD control method, is specifically what is prepared according to following steps：
Step 1: setting plane on Stewart platforms around the X-axis of Stewart platforms with angleRotated, Stewart is put down
Plane is rotated around the Y-axis of Stewart platforms with angle, θ on platform, Z axis of the plane around Stewart platforms on Stewart platforms
Rotated with angle ψ, the kinetic model such as formula of Stewart platforms is set up by Newton-Euler (Newton-Euler method) method
(1) shown in：
Wherein,For the broad sense position vector on Stewart platforms, x, y and z are respectively
Displacement of the plane on X, Y and Z axis on Stewart platforms；M∈R6×6For the inertial matrix of Stewart platforms；R is real number；B∈
R6×6For the damping matrix of Stewart platforms；K∈R6×6For the stiffness matrix of Stewart platforms；C∈R6For Stewart platforms institute
By centripetal force；For Stewart platforms coriolis acceleration vector and be χ first derivatives；Add for Stewart platforms Ke Shi
Velocity vector and for χ second dervatives；Δs∈R6For model uncertainty, Δs∈R6Do not model including parameter uncertainty and dynamic
State etc.；τ∈R6The generalized driving forces produced for the executing agency of 6 poles；ws∈R6Vector is disturbed caused by external vibration；
It is described set up Stewart platforms kinetic model specific derivation process be：
(1) vector representation of Stewart platforms is as shown in figure 1, upper mounting plate and the tie point of 6 poles are end device Ai,i
=1,2,3, the tie point of lower platform and 6 poles is Bi, i=1,2,3；{ B } is the inertial reference coordinate for being fixed on lower platform
System, its origin is overlapped with the barycenter of lower platform, and { P } is the reference frame of moving platform；rbaseBe { B } origin to pedestal connect
Point BiRadial distance, rendIt is the origin of { P } to pedestal tie point AiRadial distance；
As Fig. 1 can obtain expression formula as follows：
qi=x0+[R1]pi-ri (8)
Wherein, riIt is the tie point B represented in base coordinate systemiPosition vector, piIt is at moving platform coordinate system { P }
The end device A of middle expressioniPosition vector, x0It is moving platform barycenter C position vector, qiIt is from BiTo AiPole vector, R1
It is transition matrix of the moving platform to pedestal；
(2) the length l of poleiIt is defined as
li=| qi|=(qi Tqi)1/2 (10)
An important matrix is described below --- Jacobian matrix J, it is the length change of supporting leg and the fortune of moving platform
Dynamic contact together, can be obtained, i.e., by the principle of virtual work
Wherein, q=(q1,q2,…,q6)TThe change of strut lengths is represented,Represent upper mounting plate
Broad sense position vector；
(3) particularly, the Jacobian matrix of the Stewart platforms of cube configuration can be provided by formula (13)
Wherein, L is the length of each pole；
(4) shown in the kinetic model such as formula (14) of the Stewart platforms described by Newton-Euler methods：
The every specific definition of the above is as follows
M=Mx+JTMsJ (15)
Wherein,M is the quality of load, I ∈ R3×3For moment of inertia matrix, J ∈ R6×6For refined gram
Than matrix (Jacobian matrixes), Ms=diag ([m1, m2, m3, m4, m5, m6]), miFor the quality of i-th of movable pole；I=
1 ... 6, MsFor the quality of movable pole；τ=JTfm, fm∈R6It is the driving force produced by each pole；Here, each pole
Driving force is produced by a linear voice coil motor, according to the electromagnetic property of voice coil motor, along pole axial direction
Driving force can be expressed as fm=Kmim, wherein, Km=diag ([km1,km2,km3,km4,km5,km6])；KmFor voice coil motor torque
Coefficient matrix, im=[i1,i2,…,i6]T, imFor the current strength of coil；J=1 ..., 6, i=1 ..., 6, kmiFor voice coil motor
I-th of element in moment coefficient matrix, ijJ-th of element in counter electromotive force of motor matrix；
Wherein,I=1 ..., 6, biFor
The damped coefficient of i-th of pole；kiFor the stiffness coefficient of i-th of pole；
Step 2: the driving force of each pole is produced by a linear voice coil motor, electricity is driven according to voice coil loudspeaker voice coil
The voltage equation of machine, shown in the kinetic model such as formula (2) for six executing agencies for setting up Stewart platforms：
Wherein, L is voice coil motor inductance coefficent matrix；R is the resistor matrix of voice coil motor, KeFor the anti-electricity of voice coil motor
Kinetic potential matrix；Δm∈R6For the uncertainty of voice coil motor；U is the PD control device based on extended state observer；wm∈R6For
Interference vector caused by external vibration in motor；J is Jacobian matrix；imFor the current strength of voice coil motor coil；For im's
First derivative；
The derivation of the kinetic model of six executing agencies of Stewart platforms is as follows in described step two：
Consider the voltage equation of voice coil loudspeaker voice coil motor, the kinetic model of six executing agencies of Stewart platforms
As shown in formula (17)：
Wherein, Δm∈R6Represent the uncertainty of voice coil motor；
L=diag ([lm1,lm2,lm3,lm4,lm5,lm6]),lmi(i=1 ..., 6) represent voice coil motor inductance coefficent matrix
In i-th of element；
R=diag ([rm1, rm2, rm3, rm4, rm5, rm6]), rmi(i=1 ..., 6) is represented in the resistor matrix of voice coil motor
I-th of element；
Ke=diag ([ke1,ke2,ke3,ke4,ke5,ke6]),kei(i=1 ..., 6) represent voice coil motor counter electromotive force matrix
In i-th of element；
U=[u1,u2,…,u6]T,ui(i=1 ..., 6)；U represents control voltage, wm∈R6Expression is caused by external vibration
Interference vector, it will influence executing agency performance；
In order to simplify the research of problem, hypothesis below is done：
1), in vibration isolation application, the working space of Stewart platforms is very little, therefore, the ginseng relevant with position of platform
Number be considered it is constant, particularly, transition matrix [R1] it is considered as unit matrix I3, and centripetal force and coriolis acceleration
Item can be ignored；
2), the parameter of each pole of Stewart platforms is all identical, i.e.,：
Wherein,It is unit matrix；B is the damped coefficient of pole, and B is damped coefficient matrix, KmFor voice coil motor power
Moment coefficient matrix, msFor pole quality；keFor the element in voice coil motor counter electromotive force matrix；K is the stiffness coefficient of pole；
For real number；
3), the kinetics equation (17) of executing agency is substituted into the kinetics equation (14) of Stewart platforms, it is considered to false
If shown in the kinetics equation such as formula (18) for (1), obtaining augmentation：
By assuming 2), further to obtain
Wherein
Based on some good characteristics of cube configuration itself, the Stewart platforms of height coupling can be decoupled into 6
The passage of single-input single-output, can design the active vibration isolation controller of single-input single-output for each passage；
Note formula (19) isOrderObtain
Shown in the state-space representation of Stewart platforms such as formula (20)：
It is designated as
Wherein,
Step 3: according to the kinetic model of Stewart platforms and the dynamics of six executing agencies of Stewart platforms
Model calculates the state space for obtaining Stewart platforms；
Step 4: by the state space of Stewart platforms, design Stewart platforms extended state observer calculates expansion
State observerDetermine that extended state observer is missed to the observation error of system mode for convergent observation
Difference；
Step 5: according to the observed quantity of expansion observerThe PD control device based on extended state observer is designed,
Suppress extraneous interference disturbance to carry out the vibration isolation control of Stewart platforms using PD control device；
Wherein, kpFor the proportionality coefficient of PD control device；kdFor the differential coefficient of PD control device；x1=χ；For x2Observation
State,For x4Observer state；PD control implement body based on extended state observer is：
Present embodiment effect：
Present embodiment considers 6DOF vibration isolation problem, devises the Stewart platforms based on cube configuration
Vibration isolation algorithm, initially set up the kinematics and kinetic simulation of Stewart platforms using voice coil motor as executing agency
Type is simultaneously rationally converted to model, by the equivalent subsystem into 6 single-input single-outputs of platform, it is contemplated that the sensing in practical application
The quantity of device and the Cost Problems of equipment, devise the PD (proportion based on extended state observer
Differentiation) vibration isolation controller, and give parameter designing when considering extended state observer convergence
Criterion, designed control algolithm drastically increases the vibration isolating effect of platform while effectively reducing compared with passive vibration isolation
The quantity of required sensor.
Compared with above-mentioned algorithm, this embodiment presents the advantage that：
1. many methods all do not account for executing agency, and present embodiment establishes the dynamic of voice coil motor actuator in detail
Mechanical equation；
2. most of control method is all based on what joint space was designed, and control accuracy is limited, and present embodiment
It is controlled based on working space, control accuracy is higher；
3. present embodiment considers the immeasurability of the platform generalized velocity of the control algolithm based on workplace design, adopts
Generalized velocity information is observed with extended state observer, the complexity and cost of operating platform in practical application is reduced.
From Fig. 6~8, extended state observer can track broad sense position and the speed of system within the limited time
Signal, and the state is expanded it is estimated that external disturbance.In the case where only considering passive vibration isolation, present embodiment it is vertical
The Stewart platforms of cube configuration have good vibration isolating effect to the sinusoidal interference and random noise disturbance in broadband.Plus
Enter after the PD active vibration isolations based on extended state observer, vibration isolating effect all improves significantly, steady-state error has a quantity
The lifting of level, the especially sinusoidal interference and random noise disturbance to medium-high frequency.
Pass through the kinetic model for setting up Stewart platforms and six execution machines of Stewart platforms of present embodiment
The kinetic model process of structure can see, it is contemplated that the structural nonlinear of platform, present embodiment based on expansion state see
In the condition of convergence that provides of PD control device design for surveying device, provide specific value requirement, it is to avoid the arbitrariness of algorithm.
Embodiment two：Present embodiment from unlike embodiment one：L=diag in step 2
([lm1,lm2,lm3,lm4,lm5,lm6]), lmiFor i-th of element in voice coil motor inductance matrix；I=1 ..., 6.Other steps
And parameter is identical with embodiment one.
Embodiment three：Present embodiment from unlike embodiment one or two：R=diag in step 2
([rm1, rm2, rm3, rm4, rm5, rm6])；rmiFor i-th of element in voice coil motor resistor matrix；I=1 ..., 6.Other steps
And parameter is identical with embodiment one or two.
Embodiment four：Unlike one of present embodiment and embodiment one to three：K in step 2e
=diag ([ke1,ke2,ke3,ke4,ke5,ke6]), keiFor i-th of element in voice coil motor counter electromotive force matrix；I=1 ...,
6.Other steps and parameter are identical with one of embodiment one to three.
Embodiment five：Unlike one of present embodiment and embodiment one to four：U=in step 2
[u1,u2,…,u6]T, uiIt is i-th of element in control voltage matrix；I=1 ..., 6.Other steps and parameter and specific embodiment party
One of formula one to four is identical.
Embodiment six：Unlike one of present embodiment and embodiment one to five：Root in step 3
According to Stewart platforms kinetic model and Stewart platforms six executing agencies kinetic model calculate obtain
The detailed process of the state space of Stewart platforms is：
(1) kinetics equation (2) of executing agency is substituted into the kinetics equation (1) of Stewart platforms, obtains formula
(3)：
Wherein,
Wherein, KmFor voice coil motor moment coefficient matrix；For ΔsFirst derivative；kmFor the torque system of voice coil motor
Number；W, Δ and V are intermediate variable；K is the stiffness coefficient of pole；keFor the element in voice coil motor counter electromotive force matrix；rm=
[rm1,rm2,…,rm6], lm=[lm1,lm2,…,lm6]；
(2) note formula (3) isOrderObtain
Shown in the state-space representation of Stewart platforms such as formula (4)：
Based on some good characteristics of cube configuration itself, the Stewart platforms of height coupling can be decoupled into 6
The passage of single-input single-output, can design the active vibration isolation controller of single-input single-output for each passage；Respectively x1、
x2And x3Derivative；For χ second dervative,For χ three order derivatives；
Wherein,bs1、y1、ks1And ks2For intermediate variable.Other steps and parameter and specific implementation
One of mode one to five is identical.
Embodiment seven：Unlike one of present embodiment and embodiment one to six：In step 4 by
The state space of Stewart platforms, design Stewart platforms extended state observer calculates extended state observerIt is convergent observation error detailed process to the observation error of system mode to determine extended state observer
For：
By the state space of Stewart platforms, design Stewart platform extended state observers are as follows：
Wherein,For x1Observer state,For x2Observer state,For x3Observer state,For x4Observation shape
State；ε ＞ 0, α1,α2,α3,α4For PD control device design parameter, Stewart platform extended state observer gain alphas1、α2、α3And α4；
Multinomial s4+α1s3+α2s2+α3s+α4Meet Hurwitz conditions；
Wherein, s is the complex variable in pull-type conversion；
(2), defineObtain formula (6)：
Define error η=[η1 η2 η3 η4]T
Wherein,D is intermediate variable；D=w- Δs；
(3) and work asThen extended state observer is convergent to the observation error of system mode；
Wherein, λmin(Q) minimal eigenvalue for being Q；Q is any given positive definite symmetric matrices, exists and enables to Lyapunov equations
The positive definite symmetric matrices P of establishment；ε is extended state observer parameter；The purpose of this step is design extended state observer, and
The condition of convergence must be gone out, extended state observer is not restrained, then design process is imperfect.Other steps and parameter and specific embodiment party
One of formula one to six is identical.
Embodiment eight：Unlike one of present embodiment and embodiment one to seven：WhenThen extended state observer is convergent observation error detailed process to the observation error of system mode：
Due to
It can similarly obtain
Obtain shown in observation error state equation such as formula (26)：
Wherein, I=1 ..., 6, biFor
The damped coefficient of i-th of pole；For η first derivative；η=[η1,η2,η3,η4],For d first derivative；
Then characteristic equation is
And λ4+α1λ3+α2λ2+α3λ+α4=0 (28)
Wherein, λ representing matrixs characteristic value, I represents unit matrix；
By selecting α1、α2、α3And α4MakeFor Hurwitz, then to any given positive definite symmetric matrices Q, there is positive definite
The Lyapunov equations that symmetrical matrix P enables to formula (29) to represent are set up：
It is V to choose Lyapunov functions0=ε ηTP η, to V0=ε ηTP η derivations, are obtained：
ByThe condition of convergence for obtaining extended state observer is：
From formula (31), error η convergence speed is determined that ε is smaller by ε, and η gets over rapid convergence.Other steps and parameter with
One of embodiment one to seven is identical.
Beneficial effects of the present invention are verified using following examples：
Embodiment one：
A kind of Stewart platform active vibration isolation PD control methods based on extended state observer of the present embodiment, be specifically
Prepared according to following steps：
The present invention uses following simulation parameter：
1) load information
Quality of loads：M=12.4kg
2) platform information
The rotary inertia of upper mounting plate and payload：Isx=Isy=0.157kgm2,Isz=0.313kgm2
Each pole nominal length：L=0.2m
Each pole quality：ms=1kg
Each the damped coefficient and stiffness coefficient of pole are respectively：B=19.1kg/s, k=2000N/m
The parameter of voice coil motor：Moment coefficient km=68.9N/A, inductance lm=4.57mH, direct current generator impedance rm=6.05
Ω, back EMF coefficient ke=68.9Vs/m
3) controller design parameter
Extended state observer is High-gain observer, if extended state observer initial value and object initial value not
Together, for the ε of very little, peak phenomenon can be also produced, causes the convergence effect of extended state observer poor.In order to prevent peak value from showing
As designing following ε：
Therefore, the PD active vibration isolation controllers based on extended state observer are：
Extended state observer gain：α1=4, α2=16, α3=4, α4=1；
PD control device parameter：kp=500, kd=500；
4) simulation analysis
Simulating, verifying is carried out by choosing representational wide band sinusoidal signal interference and random noise, as a result as schemed
Shown in 3 (a)~Figure 14 (f)：
From Fig. 6 (a) -8 (f), extended state observer can track the broad sense position of system within the limited time
And rate signal, and the state is expanded it is estimated that external disturbance.In the case where only considering passive vibration isolation, this paper's is vertical
The Stewart platforms of cube configuration have good vibration isolating effect to the sinusoidal interference and random noise disturbance in broadband.Plus
Enter after the PD active vibration isolations based on extended state observer, vibration isolating effect all improves significantly, steady-state error has a quantity
The lifting of level, the especially sinusoidal interference and random noise disturbance to medium-high frequency.
The present invention can also have other various embodiments, in the case of without departing substantially from spirit of the invention and its essence, this area
Technical staff works as can make various corresponding changes and deformation according to the present invention, but these corresponding changes and deformation should all belong to
The protection domain of appended claims of the invention.
Claims (8)
1. a kind of Stewart platform active vibration isolation PD control methods based on extended state observer, it is characterised in that Yi Zhongji
In the Stewart platform active vibration isolation PD control methods of extended state observer be specifically what is followed the steps below：
Step 1: setting plane on Stewart platforms around the X-axis of Stewart platforms with angleRotated, on Stewart platforms
Plane is rotated around the Y-axis of Stewart platforms with angle, θ, on Stewart platforms plane around the Z axis of Stewart platforms with angle
Degree ψ is rotated, shown in the kinetic model such as formula (1) that Stewart platforms are set up by Newton-Euler methods：
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</mover>
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<mi>C</mi>
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<mi>&chi;</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<mi>B</mi>
<mover>
<mi>&chi;</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<mi>K</mi>
<mi>&chi;</mi>
<mo>+</mo>
<msub>
<mi>&Delta;</mi>
<mi>s</mi>
</msub>
<mo>=</mo>
<mi>&tau;</mi>
<mo>+</mo>
<msub>
<mi>w</mi>
<mi>s</mi>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein,For the broad sense position vector on Stewart platforms, x, y and z are respectively that Stewart is put down
Displacement of the plane on X, Y and Z axis on platform；M∈R6×6For the inertial matrix of Stewart platforms；R is real number；B∈R6×6For
The damping matrix of Stewart platforms；K∈R6×6For the stiffness matrix of Stewart platforms；C∈R6Suffered by Stewart platforms to
Mental and physical efforts；For Stewart platforms coriolis acceleration vector and be χ first derivatives；For Stewart platform coriolis accelerations
Vector and for χ second dervatives；Δs∈R6For model uncertainty, τ ∈ R6The broad sense driving produced for the executing agency of 6 poles
Power；ws∈R6Vector is disturbed caused by external vibration；
Step 2: according to the voltage equation of voice coil loudspeaker voice coil motor, setting up the power of six executing agencies of Stewart platforms
Learn shown in model such as formula (2)：
<mrow>
<mi>L</mi>
<msub>
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<mi>i</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>m</mi>
</msub>
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<msub>
<mi>Ri</mi>
<mi>m</mi>
</msub>
<mo>+</mo>
<msub>
<mi>K</mi>
<mi>e</mi>
</msub>
<mi>J</mi>
<mover>
<mi>&chi;</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<msub>
<mi>&Delta;</mi>
<mi>m</mi>
</msub>
<mo>=</mo>
<mi>u</mi>
<mo>+</mo>
<msub>
<mi>w</mi>
<mi>m</mi>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, L is voice coil motor inductance coefficent matrix；R is the resistor matrix of voice coil motor, KeFor the counter electromotive force of voice coil motor
Matrix；Δm∈R6For the uncertainty of voice coil motor；U is the PD control device based on extended state observer；wm∈R6For motor
Interference vector caused by middle external vibration；J is Jacobian matrix；imFor the current strength of voice coil motor coil；For imSingle order
Derivative；
Step 3: according to the kinetic model of Stewart platforms and the kinetic model of six executing agencies of Stewart platforms
Calculate the state space for obtaining Stewart platforms；
Step 4: by the state space of Stewart platforms, design Stewart platforms extended state observer calculates expansion state
ObserverIt is convergent observation error to the observation error of system mode to determine extended state observer, and η is
Error, ε is extended state observer parameter, λ representing matrix characteristic values；
Step 5: according to the observed quantity of expansion observerWithThe PD control device based on extended state observer is designed, is utilized
PD control device suppresses extraneous interference disturbance to carry out the vibration isolation control of Stewart platforms；
Wherein, kpFor the proportionality coefficient of PD control device；kdFor the differential coefficient of PD control device； For x2Observer state,For x4Observer state；PD control implement body based on extended state observer is：
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</msub>
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<mn>1</mn>
</msub>
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<msub>
<mi>K</mi>
<mi>d</mi>
</msub>
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<mi>x</mi>
<mo>^</mo>
</mover>
<mn>2</mn>
</msub>
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<mi>x</mi>
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</mover>
<mn>4</mn>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mrow>
2. a kind of Stewart platform active vibration isolation PD control methods based on extended state observer according to claim 1,
It is characterized in that：L=diag ([l in step 2m1,lm2,lm3,lm4,lm5,lm6]), lmiFor i-th in voice coil motor inductance matrix
Individual element；I=1 ..., 6.
3. a kind of Stewart platform active vibration isolation PD control methods based on extended state observer according to claim 2,
It is characterized in that：R=diag ([r in step 2m1,rm2,rm3,rm4,rm5,rm6])；rmiFor i-th in voice coil motor resistor matrix
Individual element；I=1 ..., 6.
4. a kind of Stewart platform active vibration isolation PD control methods based on extended state observer according to claim 3,
It is characterized in that：K in step 2e=diag ([ke1,ke2,ke3,ke4,ke5,ke6]), keiFor in voice coil motor counter electromotive force matrix
I-th of element；I=1 ..., 6.
5. a kind of Stewart platform active vibration isolation PD control methods based on extended state observer according to claim 4,
It is characterized in that：U=[u in step 21,u2,…,u6]T, uiIt is i-th of element in control voltage matrix；I=1 ..., 6.
6. a kind of Stewart platform active vibration isolation PD control methods based on extended state observer according to claim 5,
It is characterized in that：According to the dynamic of six executing agencies of the kinetic model of Stewart platforms and Stewart platforms in step 3
Mechanical model calculates the detailed process of state space for obtaining Stewart platforms：
(1) kinetics equation (2) of executing agency is substituted into the kinetics equation (1) of Stewart platforms, obtains formula (3)：
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<mi>bJ</mi>
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<mi>r</mi>
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<mi>l</mi>
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<mi>k</mi>
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<mi>r</mi>
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<mi>l</mi>
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</msub>
</mfrac>
<msup>
<mi>J</mi>
<mi>T</mi>
</msup>
<mi>J</mi>
<mi>&chi;</mi>
<mo>+</mo>
<mi>&Delta;</mi>
<mo>=</mo>
<mi>V</mi>
<mo>+</mo>
<mi>w</mi>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein,
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<mi>r</mi>
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<mi>l</mi>
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<mo>(</mo>
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<mi>J</mi>
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<mi>w</mi>
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<mn>1</mn>
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<mi>l</mi>
<mi>m</mi>
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<mrow>
<mo>(</mo>
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<mi>T</mi>
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<msub>
<mi>K</mi>
<mi>m</mi>
</msub>
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</mrow>
<mi>u</mi>
</mrow>
Wherein, KmFor voice coil motor moment coefficient matrix；For ΔsFirst derivative；kmFor the moment coefficient of voice coil motor；w、Δ
It is intermediate variable with V；K is the stiffness coefficient of pole；keFor the element in voice coil motor counter electromotive force matrix, b is the resistance of pole
Buddhist nun's coefficient；rm=[rm1,rm2,…,rm6], lm=[lm1,lm2,…,lm6]；
(2) note formula (3) isOrderObtain Stewart
Shown in the state-space representation of platform such as formula (4)：
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<mtd>
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<mi>x</mi>
<mo>&CenterDot;</mo>
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<mn>1</mn>
</msub>
<mo>=</mo>
<msub>
<mi>x</mi>
<mn>2</mn>
</msub>
</mrow>
</mtd>
</mtr>
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<mtd>
<mrow>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;</mo>
</mover>
<mn>2</mn>
</msub>
<mo>=</mo>
<msub>
<mi>x</mi>
<mn>3</mn>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;</mo>
</mover>
<mn>3</mn>
</msub>
<mo>=</mo>
<mo>-</mo>
<msup>
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<mn>1</mn>
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<mi>b</mi>
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<mi>s</mi>
<mn>1</mn>
</mrow>
</msub>
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<mi>x</mi>
<mn>3</mn>
</msub>
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<mi>M</mi>
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<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
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<mi>k</mi>
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<mi>s</mi>
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<mi>M</mi>
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</mrow>
</msup>
<mi>V</mi>
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<msup>
<mi>M</mi>
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<mn>1</mn>
</mrow>
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<mi>w</mi>
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<mtd>
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<msub>
<mi>y</mi>
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<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein,bs1、y1、ks1And ks2For intermediate variable.
7. a kind of Stewart platform active vibration isolation PD control methods based on extended state observer according to claim 6,
It is characterized in that：By the state space of Stewart platforms in step 4, design Stewart platforms extended state observer is calculatedIt is convergent observation error detailed process to the observation error of system mode to determine extended state observer
For：
By the state space of Stewart platforms, design Stewart platform extended state observers are as follows：
(1)、
Wherein,For x1Observer state,For x2Observer state,For x3Observer state,For x4Observer state；ε
＞ 0, α1,α2,α3,α4For PD control device design parameter, Stewart platform extended state observer gain alphas1、α2、α3And α4；
Multinomial s4+α1s3+α2s2+α3s+α4Meet Hurwitz conditions；
Wherein, s is the complex variable in pull-type conversion；
(2), defineObtain formula (6)：
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<mo>~</mo>
</mover>
<mn>4</mn>
</msub>
<mo>+</mo>
<mfrac>
<msub>
<mi>&alpha;</mi>
<mn>3</mn>
</msub>
<msup>
<mi>&epsiv;</mi>
<mn>3</mn>
</msup>
</mfrac>
<msub>
<mover>
<mi>x</mi>
<mo>~</mo>
</mover>
<mn>1</mn>
</msub>
<mo>-</mo>
<msup>
<mi>M</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<msub>
<mi>b</mi>
<mrow>
<mi>s</mi>
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mn>3</mn>
</msub>
<mo>-</mo>
<msup>
<mi>M</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<msub>
<mi>k</mi>
<mrow>
<mi>s</mi>
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mn>2</mn>
</msub>
<mo>-</mo>
<msup>
<mi>M</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<msub>
<mi>k</mi>
<mrow>
<mi>s</mi>
<mn>2</mn>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>B</mi>
<mi>v</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mover>
<mi>x</mi>
<mo>~</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mn>4</mn>
</msub>
<mo>=</mo>
<mfrac>
<msub>
<mi>&alpha;</mi>
<mn>4</mn>
</msub>
<msup>
<mi>&epsiv;</mi>
<mn>4</mn>
</msup>
</mfrac>
<msub>
<mover>
<mi>x</mi>
<mo>~</mo>
</mover>
<mn>1</mn>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
Define error η=[η1 η2 η3 η4]T
Wherein,D is intermediate variable；D=w- Δs；
(3) and work asThen extended state observer is convergent to the observation error of system mode；Wherein,
λmin(Q) minimal eigenvalue for being Q；Q is any given positive definite symmetric matrices, exists and enables to Lyapunov equations to set up
Positive definite symmetric matrices P；ε is extended state observer parameter.
8. a kind of Stewart platform active vibration isolation PD control methods based on extended state observer according to claim 7,
It is characterized in that：WhenThen extended state observer is missed to the observation error of system mode for convergent observation
Poor detailed process：
Due to
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<mi>&epsiv;</mi>
<msub>
<mover>
<mi>&eta;</mi>
<mo>&CenterDot;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>-</mo>
<msub>
<mover>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mn>1</mn>
</msub>
</mrow>
<msup>
<mi>&epsiv;</mi>
<mn>2</mn>
</msup>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<msup>
<mi>&epsiv;</mi>
<mn>2</mn>
</msup>
</mfrac>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<mn>2</mn>
</msub>
<mo>-</mo>
<mo>(</mo>
<mrow>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mn>2</mn>
</msub>
<mo>+</mo>
<mfrac>
<msub>
<mi>&alpha;</mi>
<mn>1</mn>
</msub>
<mi>&epsiv;</mi>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>y</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mn>1</mn>
</msub>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<msup>
<mi>&epsiv;</mi>
<mn>2</mn>
</msup>
</mfrac>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<mn>2</mn>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mn>2</mn>
</msub>
<mo>-</mo>
<mfrac>
<msub>
<mi>&alpha;</mi>
<mn>1</mn>
</msub>
<mi>&epsiv;</mi>
</mfrac>
<mo>(</mo>
<mrow>
<msub>
<mi>y</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mn>1</mn>
</msub>
</mrow>
<mo>)</mo>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<msub>
<mi>&alpha;</mi>
<mn>1</mn>
</msub>
<msup>
<mi>&epsiv;</mi>
<mn>3</mn>
</msup>
</mfrac>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<msup>
<mi>&epsiv;</mi>
<mn>2</mn>
</msup>
</mfrac>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<mn>2</mn>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>x</mi>
<mo>^</mo>
</mover>
<mn>2</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>&alpha;</mi>
<mn>1</mn>
</msub>
<msub>
<mi>&eta;</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msub>
<mi>&eta;</mi>
<mn>2</mn>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>24</mn>
<mo>)</mo>
</mrow>
</mrow>
It can similarly obtain
<mrow>
<mi>&epsiv;</mi>
<msub>
<mover>
<mi>&eta;</mi>
<mo>&CenterDot;</mo>
</mover>
<mn>2</mn>
</msub>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>&alpha;</mi>
<mn>2</mn>
</msub>
<msub>
<mi>&eta;</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msub>
<mi>&eta;</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<mi>&epsiv;</mi>
<msub>
<mover>
<mi>&eta;</mi>
<mo>&CenterDot;</mo>
</mover>
<mn>3</mn>
</msub>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>&alpha;</mi>
<mn>3</mn>
</msub>
<msub>
<mi>&eta;</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msub>
<mi>&eta;</mi>
<mn>4</mn>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>25</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>&epsiv;</mi>
<msub>
<mover>
<mi>&eta;</mi>
<mo>&CenterDot;</mo>
</mover>
<mn>4</mn>
</msub>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>&alpha;</mi>
<mn>4</mn>
</msub>
<msub>
<mi>&eta;</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>&epsiv;</mi>
<mover>
<mi>d</mi>
<mo>&CenterDot;</mo>
</mover>
</mrow>
Obtain shown in observation error state equation such as formula (26)：
<mrow>
<mi>&epsiv;</mi>
<mover>
<mi>&eta;</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>=</mo>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mi>&eta;</mi>
<mo>+</mo>
<mi>&epsiv;</mi>
<mover>
<mi>B</mi>
<mo>&OverBar;</mo>
</mover>
<mover>
<mi>d</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>26</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein,I=1 ..., 6, biFor
The damped coefficient of i pole；For η first derivative；η=[η1,η2,η3,η4],For d first derivative；
Then characteristic equation is
<mrow>
<mrow>
<mo>|</mo>
<mi>&lambda;I</mi>
<mo>-</mo>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mfenced open='|' close='|'>
<mtable>
<mtr>
<mtd>
<mi>&lambda;</mi>
</mtd>
<mtd>
<mo>-</mo>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>&alpha;</mi>
<mn>2</mn>
</msub>
</mtd>
<mtd>
<mi>&lambda;</mi>
</mtd>
<mtd>
<mo>-</mo>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>&alpha;</mi>
<mn>3</mn>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mi>&lambda;</mi>
</mtd>
<mtd>
<mo>-</mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>&alpha;</mi>
<mn>4</mn>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mi>&lambda;</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>27</mn>
<mo>)</mo>
</mrow>
</mrow>
And λ4+α1λ3+α2λ2+α3λ+α4=0 (28)
Wherein, λ representing matrixs characteristic value, I represents unit matrix；
By selecting α1、α2、α3And α4MakeFor Hurwitz, then to any given positive definite symmetric matrices Q, there is positive definite symmetrical
The Lyapunov equations that matrix P enables to formula (29) to represent are set up：
<mrow>
<msup>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
<mi>P</mi>
<mo>+</mo>
<mi>P</mi>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mo>+</mo>
<mi>Q</mi>
<mo>=</mo>
<mn>0</mn>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>29</mn>
<mo>)</mo>
</mrow>
</mrow>
It is V to choose Lyapunov functions0=ε ηTP η, to V0=ε ηTP η derivations, are obtained
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>V</mi>
<mo>&CenterDot;</mo>
</mover>
<mn>0</mn>
</msub>
<mo>=</mo>
<mi>&epsiv;</mi>
<msup>
<mover>
<mi>&eta;</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>T</mi>
</msup>
<mi>P</mi>
<mi>&eta;</mi>
<mo>+</mo>
<msup>
<mi>&epsiv;&eta;</mi>
<mi>T</mi>
</msup>
<mi>P</mi>
<mover>
<mi>&eta;</mi>
<mo>&CenterDot;</mo>
</mover>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mi>&eta;</mi>
<mo>+</mo>
<mi>&epsiv;</mi>
<mover>
<mi>B</mi>
<mo>&OverBar;</mo>
</mover>
<mover>
<mi>d</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<mi>P</mi>
<mi>&eta;</mi>
<mo>+</mo>
<msup>
<mi>&eta;</mi>
<mi>T</mi>
</msup>
<mi>P</mi>
<mrow>
<mo>(</mo>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mi>&eta;</mi>
<mo>+</mo>
<mi>&epsiv;</mi>
<mover>
<mi>B</mi>
<mo>&OverBar;</mo>
</mover>
<mover>
<mi>d</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msup>
<mi>&eta;</mi>
<mi>T</mi>
</msup>
<msup>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
<mi>P</mi>
<mi>&eta;</mi>
<mo>+</mo>
<mi>&epsiv;</mi>
<msup>
<mrow>
<mo>(</mo>
<mover>
<mi>B</mi>
<mo>&OverBar;</mo>
</mover>
<mover>
<mi>d</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<mi>P</mi>
<mi>&eta;</mi>
<mo>+</mo>
<msup>
<mi>&eta;</mi>
<mi>T</mi>
</msup>
<mi>P</mi>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mi>&eta;</mi>
<mo>+</mo>
<msup>
<mi>&epsiv;&eta;</mi>
<mi>T</mi>
</msup>
<mi>P</mi>
<mover>
<mi>B</mi>
<mo>&OverBar;</mo>
</mover>
<mover>
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<mo>&CenterDot;</mo>
</mover>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msup>
<mi>&eta;</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<msup>
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<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mi>T</mi>
</msup>
<mi>P</mi>
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<mi>P</mi>
<mover>
<mi>A</mi>
<mo>&OverBar;</mo>
</mover>
<mo>)</mo>
</mrow>
<mi>&eta;</mi>
<mo>+</mo>
<mn>2</mn>
<msup>
<mi>&epsiv;&eta;</mi>
<mi>T</mi>
</msup>
<mi>P</mi>
<mover>
<mi>B</mi>
<mo>&OverBar;</mo>
</mover>
<mover>
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<mo>&CenterDot;</mo>
</mover>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>&le;</mo>
<mo>-</mo>
<msup>
<mi>&eta;</mi>
<mi>T</mi>
</msup>
<mi>Q</mi>
<mi>&eta;</mi>
<mo>+</mo>
<mn>2</mn>
<mi>&epsiv;</mi>
<mo>|</mo>
<mo>|</mo>
<mi>P</mi>
<mover>
<mi>B</mi>
<mo>&OverBar;</mo>
</mover>
<mo>|</mo>
<mo>|</mo>
<mo>&CenterDot;</mo>
<mo>|</mo>
<mo>|</mo>
<mi>&eta;</mi>
<mo>|</mo>
<mo>|</mo>
<mo>&CenterDot;</mo>
<mo>|</mo>
<mover>
<mi>d</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>|</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>&le;</mo>
<mo>-</mo>
<msub>
<mi>&lambda;</mi>
<mi>min</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>Q</mi>
<mo>)</mo>
</mrow>
<mo>|</mo>
<mo>|</mo>
<mi>&eta;</mi>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
<mo>+</mo>
<mn>2</mn>
<mi>&epsiv;</mi>
<mi>L</mi>
<mo>|</mo>
<mo>|</mo>
<mi>P</mi>
<mover>
<mi>B</mi>
<mo>&OverBar;</mo>
</mover>
<mo>|</mo>
<mo>|</mo>
<mo>|</mo>
<mo>|</mo>
<mi>&eta;</mi>
<mo>|</mo>
<mo>|</mo>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>30</mn>
<mo>)</mo>
</mrow>
</mrow>
ByThe condition of convergence for obtaining extended state observer is：
<mrow>
<mo>|</mo>
<mo>|</mo>
<mi>&eta;</mi>
<mo>|</mo>
<mo>|</mo>
<mo>&le;</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>&epsiv;</mi>
<mi>L</mi>
<mo>|</mo>
<mo>|</mo>
<mi>P</mi>
<mover>
<mi>B</mi>
<mo>&OverBar;</mo>
</mover>
<mo>|</mo>
<mo>|</mo>
</mrow>
<mrow>
<msub>
<mi>&lambda;</mi>
<mi>min</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>Q</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>31</mn>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mrow>
4
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