CN107703750A - A kind of networking multiaxial motion position synchronization control method based on automatic disturbance rejection controller - Google Patents
A kind of networking multiaxial motion position synchronization control method based on automatic disturbance rejection controller Download PDFInfo
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Abstract
A kind of networking multiaxial motion position synchronization control method based on automatic disturbance rejection controller, a part of this method first against each single-axis servo system by Uncertainty Management caused by time delay for the disturbance of system summation, and then linear extended state observer is designed, it is estimated.Secondly, synchronous error model is established, and then designs the PD type position synchronization control devices with disturbance compensation function, while the system of realization has good uniaxiality tracking control performance, realizes good position synchronization control performance.The present invention ensures that system has good interference free performance and robust performance while can effectively handle time-vary delay system to networking multiaxial motion position synchronization control systematic influence.
Description
Technical field
The present invention be applied to networking motion control field, be related to it is a kind of suitable for networking Multi-axis coordinated motion control
Position synchronization control method.
Background technology
In modern intelligence manufacture industry, the application of Multi-axis motion control is increasingly extensive, can be achieved again by multi-shaft interlocked
Miscellaneous functions of the equipments, such as industrial robot, shaftless printing press, weaving loom and printing packaging machine.With the quick hair of network technology
Exhibition, multi-shaft motion control system just develop towards the direction of networking and high speed.Network is introduced into servo-control system, controlled
Row data communication is entered by Ethernet between device and multiple-axis servo driver processed, substantially increased between controller and driver
Message transmission rate and reliability, while accurate multi-axial Simultaneous function is also achieved, system wiring is greatly reduced, is improved
System extended capability.Commodity ethernet has the incomparable advantage of fieldbus, base in bandwidth, cost and opening etc.
Flexibility, rapidity and the control accuracy of equipment can be improved well in the servo-drive system of the commodity ethernet development of opening.
Therefore, the Multi-axis motion control based on commodity ethernet has been increasingly becoming one of core technology of modern intelligence manufacture.
However, Ethernet is designed for commercial situations, Ethernet introducing kinetic control system is added new
Factor and problem, for example, Ethernet uncertainty communication mechanism causes various communication uncertain problems, it is difficult to meet modern sport
Control system hard real-time and high speed, high-precision processing request.Although some existing commercial industrial ethernet technologys, such as
EtherCAT, SERCOS-III, POWERLINK, but be to realize that deterministic data passes by changing data link layer protocol mostly
It is defeated.Therefore, these commercial Ethernets may be considered a kind of high-speed field bus, it is necessary to which special chip realizes protocol stack, special
Develop software and carry out system development, cost is high, technical licensing is difficult and incompatible standard ethernet.If it can propose to solve from control plane
Certainly Ethernet information transfer uncertainty will have great theory significance to the theory and method of kinetic control system performance impact
And actual application value.Meanwhile realize that multiaxial motion position synchronization control is a core skill in Multi-axis coordinated motion control
Art, it is related to single shaft position servo control and multiaxis position synchronization control.The main target of position servo control be improve position with
Track precision and interference free performance, people also been proposed many advanced control methods, PID control, sliding formwork control such as with feedforward
System, Self Adaptive Control and fuzzy control.Although the transfer rate of real-time ethernet is significantly improved, network lures
The influence of sampling dithering that time delay is brought to position tracking precision is led still to can not ignore, existing position servo control method and
Position synchronization control method is rare to consider these influences.In many network inducement delay compensation of network control system field
Method, such as PREDICTIVE CONTROL, self-adapting Smith predictor, but But most of algorithms is more complicated, is not suitable for commercial Application.Recently, it is Japanese
Scholar Natori and Ohnishi propose Communication Jamming observer (CDOB), network inducement delay are modeled as disturbing, and pass through
Communication Jamming observer carries out real-time estimation compensation, obtains good delay compensation effect, but needs system accurate model, has
Certain limitation.At present, there is no good solution still for the position synchronization control problem of networking multiaxial motion.
The content of the invention
In order to overcome the shortcomings of the position synchronization control method in existing network multi-shaft motion control system, the present invention carries
A kind of networking multiaxial motion position synchronization control method based on automatic disturbance rejection controller is supplied.First, by caused by time delay not
The part that certainty processing disturbs for system summation, and then linear extended state observer (LESO) is designed, in estimating system
While state, summation disturbance is estimated.Secondly, synchronous error model, PD of the design with disturbance compensation function are established
Type position synchronization control device, while the system of realization has good uniaxiality tracking control performance, realize that good position is same
Walk control performance.
In order to solve the above-mentioned technical problem the technical solution adopted by the present invention is as follows：
A kind of networking multiaxial motion position synchronization control method based on automatic disturbance rejection controller, methods described include as follows
Step：
For step 1) in the case where network inducement delay is less than a sampling period, foundation contains time-varying network inducing delay
Single-axis servo control system model, by networking single-axis servo control system modelling for one with a step input delay from
Dissipate linearly time-varying system, and then one that the uncertain dynamic processing of system caused by time-vary delay system is disturbed for the summation of system
Part, including procedure below：
1.1) Multi-asis servo system state-space model is established：
According to the dynamic characteristic of multi-shaft motion control system, (i=1 ..., n) the axle servo-drive system i-th under velocity mode is obtained
State-space model be
Wherein, xi1And x (t)i2(t) position quantity and speed amount of the i-th axle servo-drive system are represented respectively,For i-th
Axle servo system control inputs, i.e. speed setting value,For the i-th axle servo-drive system is unknown and the interference volume of bounded,It is position quantity for the i-th axle servo-drive system output valve, ai、biFor the model constant coefficient of the i-th axle servo-drive system；
1.2) the single-axis servo control system model established under the influence of time-varying network inducing delay：
There is network inducement delay in packet, use in network transmission processRepresent that the i-th axle sensor to controller leads to
The time delay in road,The i-th axis controller is represented to the time delay of actuator channels, then in systematic sampling moment tkUnknown time-varying be present
Network inducement delay, it is designated as Due to sensor node using the time drive, controller node and
Actuator node is event-driven, network delayA respectively less than sampling period T, then in any sampling period (tk,tk+1]
Interior, the control input for acting on actuator is made up of two parts, and a part is that the control being calculated by a upper controlling cycle is defeated
Enter ui(k-1), another part is the control input u that current control period is calculatedi(k), and form represents as follows：
Therefore, according to formula (1) and (2), using the single-axis servo after sampling period T discretization control system model as：
WillUse 1-aiAfter T approximations, formula (3) is turned to：
By in formula (4) by time-vary delay systemCaused time-varying dynamic and system interference are considered as the disturbance of system summation, with one
New variable xi3(k) represent, i.e.,
And makeThus the networking single-axis servo that will be represented by formula (4) controls system mould
Type expands into following third-order system model：
Wherein, xi1(k+1)、xi2(k+1)、xi3(k+1) represent that the i-th axle servo-drive system position exports x respectivelyi1(k), motor
Speed xi2(k), new expansion state amount xi3(k) in the value of+1 sampling instant of kth；
Step 2) establishes networking multiaxial motion position synchronization coupling error model；
Isochronous controller of the step 3) design based on linear active disturbance rejection control, realizes that networking multiaxial motion position is synchronously controlled
System.
Further, the step 2), establishes networking multiaxial motion position synchronization coupling error model, and process is as follows：
2.1) defining multiaxis position synchronous error model is
ε (k)=Γ e (k) (6)
Wherein, ε (k), e (k) are respectively multiaxis position synchronous error vector sum multiaxis position error vector, and ε (k)=
[ε1(k),…,εi(k),…,εn(k)] ', e (k)=[e11(k),…,ei1(k),…,en1(k)] ', symbol " ' " representing matrix
Transposition, εi(k)、ei1(k) the i-th (i=1 ..., n) shaft position synchronous error and site error are represented respectively, and Γ represents synchronous conversion
Matrix；
The synchronous transformation matrix Γ of selection is as follows：
I.e. position synchronous error represents as follows：
2.2) establishing multiaxis position synchronization coupling error model is
E (k)=e (k)+α ε (k) (9)
Wherein, E (k)=[E1(k),…,Ei(k),…,En(k)], and α is diagonal and positive definite control gain matrix, will
Formula (6) substitutes into (9) and obtained
E (k)=(I+ α Γ) e (k) (10)
Wherein, I represent unit matrix, when (I+ α Γ) can the inverse time, E (k) → 0 can release e (k) → 0, further, by e
(k) ε (k) → 0 → 0 is released.
Further, in the step 3), the process for designing the isochronous controller based on linear active disturbance rejection control is as follows：
3.1) to the i-th axle Servo Control System Design linearity tracking differentiator, i=1 ..., n, transition process arranging, give
Signal v0As reference input, position v is obtained by Nonlinear Tracking Differentiator0Approximate velocity differential signal vi2(k), obtain simultaneously
Setting signal v0Transition value vi1(k), the reference signal of setting is smoothed, prevents excessive overshoot, its form is as follows：
Wherein, ri0For the Turbo Factor of Nonlinear Tracking Differentiator, fhi(k) it is vi2Differential value, T is the sampling period；
3.2) to the i-th axle Servo Control System Design linear extended state observer, state and summation disturbance to system
(uncertain and Unmarried pregnancy etc. as caused by time delay) carries out estimation in real time and compensation, its form are as follows：
Wherein, ei(k) it is the i-th axle servo-drive system physical location and the difference of its estimate, i.e. the position margin of error, zi1(k) it is
To the i-th axle servo-drive system position xi1(k) estimation, zi2(k) it is to speed xi2(k) estimation, zi3(k) it is to new expansion state
Amount is summation disturbance xi3(k) estimation, βi1、βi2、βi3For one group of parameter to be adjusted, to ensure estimated accuracy, according to high-gain
State Observer Design principle, βi1、βi2、βi3Value is more than the upper bound of noise or disturbance, and can take β by POLE PLACEMENT USINGi1=3
ωi0, βi2=3 ωi0 2,ωi0For observer bandwidth, bi0For adjustable compensating factor；
3.3) for the uncertain and summation disturbance as caused by time-vary delay system in compensation system, in i-th (i=1 ..., n)
The controlled quentity controlled variable u that axle servo-drive system obtainsi0(k) z is subtracted ini3(k) new controlled quentity controlled variable, i.e. u are obtainedi(k)=ui0(k)-zi3(k)/
bi0, summation disturbance caused by time-vary delay system can be included in bucking-out system, eliminate influence of the time-vary delay system to systematic function, design
The following Synchronization Control with disturbance compensation is restrained：
Wherein, Kp、KdAnd KeTo control gain matrix, e (k)=[e11(k) … ei1(k) … en1(k)],ei1(k) it is the Setting signal transition value v of the i-th axle servo-drive systemi1(k) with
Position estimation value zi1(k) error, ei2(k) it is the differential value v of the i-th axle servo-drive system Setting signali2(k) with velocity estimation value
vi2(k) error, ui0(k) it is synchronous error feedback control amount, ui(k) it is final controlled quentity controlled variable.
Compared with prior art, the beneficial effects of the present invention are：It is to be by Uncertainty Management caused by time-vary delay system
A part for the summation disturbance of system, and linear extended state observer (LESO) is designed, while estimating system state, to total
Estimated with disturbance, and then design the PD types Synchronization Control rule with compensation interference and realize the synchronization of networking multiaxial motion position
Control.This programme can effectively handle influence of the time-vary delay system to system, while have good robust performance, and controlling party
The design of method need to only obtain the order of system, without system accurate model, can be generalized to well in sector application.
Brief description of the drawings
Fig. 1 is the position synchronization control structure chart based on automatic disturbance rejection controller.
Fig. 2 is the sampling period sequence and time-delay series figure of experimental verification.
Fig. 3 is the position synchronization control design sketch of experimental verification.
Fig. 4 is the site error design sketch of experimental verification.
Fig. 5 is the position coupling synchronous error figure of experimental verification.
Fig. 6 is each axle interference estimate of experimental verification.
Embodiment
In order that technical scheme, mentality of designing can become apparent from, retouched in detail again below in conjunction with the accompanying drawings
State.
1~Fig. 6 of reference picture, a kind of networking multiaxial motion position synchronization control method based on automatic disturbance rejection controller, institute
The method of stating comprises the following steps：
For step 1) in the case where network inducement delay is less than a sampling period, foundation contains time-varying network inducing delay
Single-axis servo control system model, by networking single-axis servo control system modelling for one with a step input delay from
Dissipate linearly time-varying system, and then one that the uncertain dynamic processing of system caused by time-vary delay system is disturbed for the summation of system
Part, including procedure below：
1.1) Multi-asis servo system state-space model is established：
According to the dynamic characteristic of multi-shaft motion control system, (i=1 ..., n) the axle servo-drive system i-th under velocity mode is obtained
State-space model be
Wherein, xi1And x (t)i2(t) position quantity and speed amount of the i-th axle servo-drive system are represented respectively,For i-th
Axle servo system control inputs, i.e. speed setting value,For the i-th axle servo-drive system is unknown and the interference volume of bounded,It is position quantity for the i-th axle servo-drive system output valve, ai、biFor the model constant coefficient of the i-th axle servo-drive system；
1.2) the single-axis servo control system model established under the influence of time-varying network inducing delay：
There is network inducement delay in packet, use in network transmission processRepresent that the i-th axle sensor to controller leads to
The time delay in road,The i-th axis controller is represented to the time delay of actuator channels, then in systematic sampling moment tkUnknown time-varying be present
Network inducement delay, it is designated as Due to sensor node using the time drive, controller node and
Actuator node is event-driven, network delayA respectively less than sampling period T, then in any sampling period (tk,tk+1]
Interior, the control input for acting on actuator is made up of two parts, and a part is that the control being calculated by a upper controlling cycle is defeated
Enter ui(k-1), another part is the control input u that current control period is calculatedi(k), and form represents as follows：
Therefore, according to formula (1) and (2), using single-axis servo after sampling period T discretization control system model as：
WillUse 1-aiAfter T approximations, formula (3) can be turned to：
By in formula (4) by time-vary delay systemCaused time-varying dynamic and system interference are considered as the disturbance of system summation, with one
New variable xi3(k) represent, i.e.,
And makeThus the networking single-axis servo that will be represented by formula (4) controls system mould
Type expands into following third-order system model：
Wherein, xi1(k+1)、xi2(k+1)、xi3(k+1) represent that the i-th axle servo-drive system position exports x respectivelyi1(k), motor
Speed xi2(k), new expansion state amount xi3(k) in the value of+1 sampling instant of kth；
Step 2) establishes networking multiaxial motion position synchronization coupling error model；
Isochronous controller of the step 3) design based on linear active disturbance rejection control, realizes that networking multiaxial motion position is synchronously controlled
System.
Further, in the step 2), the process for establishing networking multiaxial motion position synchronization coupling error model is as follows：
2.1) defining multiaxis position synchronous error model is
ε (k)=Γ e (k) (6)
Wherein, ε (k), e (k) are respectively multiaxis position synchronous error vector sum multiaxis position error vector, and ε (k)=
[ε1(k),…,εi(k),…,εn(k)] ', e (k)=[e11(k),…,ei1(k),…,en1(k)] ', symbol " ' " representing matrix
Transposition, εi(k)、ei1(k) the i-th (i=1 ..., n) shaft position synchronous error and site error are represented respectively, and Γ represents synchronous conversion
Matrix；
The synchronous transformation matrix Γ of selection is as follows：
I.e. position synchronous error represents as follows：
2.2) establishing multiaxis position synchronization coupling error model is
E (k)=e (k)+α ε (k) (9)
Wherein, E (k)=[E1(k),…,Ei(k),…,En(k)], and α is diagonal and positive definite control gain matrix, will
Formula (6) substitutes into (9) and obtained
E (k)=(I+ α Γ) e (k) (10)
Wherein, I represent unit matrix, when (I+ α Γ) can the inverse time, E (k) → 0 can release e (k) → 0, further, by e
(k) ε (k) → 0 can → 0 be released.
Further, in the step 3), the process for designing the isochronous controller based on linear active disturbance rejection control is as follows：
3.1) to the i-th axle Servo Control System Design linearity tracking differentiator, i=1 ..., n, transition process arranging, give
Signal v0As reference input, position v is obtained by Nonlinear Tracking Differentiator0Approximate velocity differential signal vi2(k), obtain simultaneously
Setting signal v0Transition value vi1(k), the reference signal of setting is smoothed, prevents excessive overshoot, its form is as follows：
Wherein, ri0For the Turbo Factor of Nonlinear Tracking Differentiator, fhi(k) it is vi2Differential value, T is the sampling period；
3.2) to the i-th axle Servo Control System Design linear extended state observer, state and summation disturbance to system
(uncertain and Unmarried pregnancy etc. as caused by time delay) carries out estimation in real time and compensation, its form are as follows：
Wherein, ei(k) it is the i-th axle servo-drive system physical location and the difference of its estimate, i.e. the position margin of error, zi1(k) it is
To the i-th axle servo-drive system position xi1(k) estimation, zi2(k) it is to speed xi2(k) estimation, zi3(k) it is to new expansion state
Amount is summation disturbance xi3(k) estimation, βi1、βi2、βi3For one group of parameter to be adjusted, to ensure estimated accuracy, according to high-gain
State Observer Design principle, βi1、βi2、βi3Value is generally higher than the upper bound of noise or disturbance, and can take β by POLE PLACEMENT USINGi1
=3 ωi0, βi2=3 ωi0 2,ωi0For observer bandwidth, bi0For adjustable compensating factor；
3.3) for the uncertain and summation disturbance as caused by time-vary delay system in compensation system, in i-th (i=1 ..., n)
The controlled quentity controlled variable u that axle servo-drive system obtainsi0(k) z is subtracted ini3(k) new controlled quentity controlled variable, i.e. u are obtainedi(k)=ui0(k)-zi3(k)/
bi0, compensation process can include summation caused by time-vary delay system in bucking-out system and disturb, and eliminate time-vary delay system to systematic function
Influence, the Synchronization Control with disturbance compensation is restrained as follows for design：
Wherein, Kp、KdAnd KeTo control gain matrix, e (k)=[e11(k)…ei1(k)…en1(k)],ei1(k) it is the Setting signal transition value v of the i-th axle servo-drive systemi1(k) with
Position estimation value zi1(k) error, ei2(k) it is the differential value v of the i-th axle servo-drive system Setting signali2(k) with velocity estimation value
vi2(k) error, ui0(k) it is synchronous error feedback control amount, ui(k) it is final controlled quentity controlled variable.
For the validity and superiority of checking institute extracting method, the present invention is carried out on the networking Motion Control Platform of four axles
Experimental verification, sets primary condition and partial parameters in experiment, i.e., in position synchronization control device based on Active Disturbance Rejection Control
The parameter difference of selection is as follows：KP=[0.02 0.02 0.02 0.02], Kd=[3.4 3.4 3.4 3.4], Ke=[2 22
2], r1=r2=r3=r3=100, b01=b02=b03=b04=18, ω10=ω20=ω30=ω40=100, α=diag
{ 0.5,0.5,0.5,0.5 }.
Sampling period is set as T=5ms, and sampling period sequence and time-delay series are as shown in Figure 2.Fig. 3 and Fig. 4 is real respectively
The position synchronization control effect and site error effect of research are tested, Fig. 5 and Fig. 6 are respectively that coupling synchronous error and the interference of each axle are estimated
Evaluation.As seen in figures 3-5, synchronously controlled using the networking multiaxial motion position of the present invention based on automatic disturbance rejection controller
Method processed, even if time-varying network inducement delay be present, the position output of multi-shaft motion control system still has response quickly
Speed, without overshoot, and steady-state error very little is almost nil, illustrates that uncertain dynamic energy is effective as caused by network inducement delay
Ground is compensated, and multi-shaft motion control system is not influenceed by time-vary delay system on its performance substantially.When multi-axial Simultaneous motion control system
System introduces an interference volume at the 7.8s moment in the 4th axle, the 4th axle is produced 10mm site error, more as we can see from the figure
The position of axle kinetic control system soon tracks given reference value, and can soon tend to be synchronous, and position is synchronously missed
Poor very little, without there is obvious shake after reference value is reached, when there is external disturbance still can effective compensation time-varying network lure
Lead and dynamic is not known caused by time delay, show that system has good synchronization and tracking control performance.In summary, designed base
Not only there is good compensation effect to network inducement delay in the synchronization control algorithm of automatic disturbance rejection controller, and outside is made an uproar
Sound also has good rejection ability.
Described above is the experimental result that the present invention provides, and fully shows the superiority of designed method, it is clear that this hair
It is bright to be not limited only to examples detailed above, without departing from its general principles and without departing from scope involved by substantive content of the present invention
On the premise of, a variety of deformations can be made to it and be carried out.Scheme designed by the present invention can effectively solve the problem that networking multiaxis is transported
The position synchronization control problem of autocontrol system, time-vary delay system can be effectively handled to networking multiaxial motion position synchronization control
While systematic influence, ensure that system has good interference free performance and robust performance.
Claims (3)
- A kind of 1. networking multiaxial motion position synchronization control method based on automatic disturbance rejection controller, it is characterised in that：The side Method comprises the following steps：Step 1) establishes the list containing time-varying network inducing delay in the case where network inducement delay is less than a sampling period Axle servo-control system model, by networking single-axis servo control system modelling for one have a step input delay it is discrete when Between linear time varying system, and then one that the uncertain dynamic processing of system caused by time-vary delay system is disturbed for the summation of system Point, including procedure below：1.1) Multi-asis servo system state-space model is established：According to the dynamic characteristic of multi-shaft motion control system, the shape of (i=1 ..., n) axle servo-drive system i-th under velocity mode is obtained State space model is<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>u</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>w</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>Wherein, xi1And x (t)i2(t) position quantity and speed amount of the i-th axle servo-drive system are represented respectively,Watched for the i-th axle Dress system control input, i.e. speed setting value,For the i-th axle servo-drive system is unknown and the interference volume of bounded,It is position quantity for the i-th axle servo-drive system output valve, ai、biFor the model constant coefficient of the i-th axle servo-drive system；1.2) the single-axis servo control system model established under the influence of time-varying network inducing delay：There is network inducement delay in packet, use in network transmission processRepresent the i-th axle sensor to controller channel Time delay,The i-th axis controller is represented to the time delay of actuator channels, then in systematic sampling moment tkUnknown time-varying network be present Inducing delay, it is designated as Because sensor node is using time driving, controller node and execution Device node is event-driven, network delayA respectively less than sampling period T, then in any sampling period (tk,tk+1] in, make Control input used in actuator is made up of two parts, and a part is the control input u being calculated by a upper controlling cyclei (k-1), another part is the control input u that current control period is calculatedi(k), form represents as follows：<mrow> <msub> <mi>u</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>u</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>t</mi> <mo>&Element;</mo> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>,</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>+</mo> <msubsup> <mi>&tau;</mi> <mi>k</mi> <mi>i</mi> </msubsup> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>u</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>t</mi> <mo>&Element;</mo> <mo>(</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>+</mo> <msubsup> <mi>&tau;</mi> <mi>k</mi> <mi>i</mi> </msubsup> <mo>,</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>Therefore, according to formula (1) and (2), using the single-axis servo after sampling period T discretization control system model as：<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Tx</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>T</mi> </mrow> </msup> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>a</mi> <mi>i</mi> </msub> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msubsup> <mi>&tau;</mi> <mi>k</mi> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>-</mo> <msub> <mi>w</mi> <mi>i</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>a</mi> <mi>i</mi> </msub> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>T</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msubsup> <mi>&tau;</mi> <mi>k</mi> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>-</mo> <msub> <mi>w</mi> <mi>i</mi> </msub> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>WillUse 1-aiAfter T approximations, formula (3) is turned to：<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Tx</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <mo>&lsqb;</mo> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>T</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msubsup> <mi>&tau;</mi> <mi>k</mi> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>-</mo> <msub> <mi>w</mi> <mi>i</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>T</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>T</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msubsup> <mi>&tau;</mi> <mi>k</mi> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>-</mo> <msub> <mi>w</mi> <mi>i</mi> </msub> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>By in formula (4) by time-vary delay systemCaused time-varying dynamic and system interference are considered as the disturbance of system summation, with a new change Measure xi3(k) represent, i.e.,<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>3</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <msub> <mi>b</mi> <mi>i</mi> </msub> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>T</mi> </mrow> </mfrac> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msubsup> <mi>&tau;</mi> <mi>k</mi> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </msup> <msub> <mi>u</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>T</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msubsup> <mi>&tau;</mi> <mi>k</mi> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>w</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>T</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>T</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>T</mi> <mo>-</mo> <msubsup> <mi>&tau;</mi> <mi>k</mi> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msub> <mi>b</mi> <mi>i</mi> </msub> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>-</mo> <msub> <mi>w</mi> <mi>i</mi> </msub> <mo>(</mo> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>And makeThus will be expanded by the networking single-axis servo control system model that formula (4) represents Open into following third-order system model：<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Tx</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>3</mn> </mrow> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>+</mo> <mfrac> <msub> <mi>b</mi> <mi>i</mi> </msub> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mi>T</mi> </mrow> </mfrac> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>3</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mn>3</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Td</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>Wherein, xi1(k+1)、xi2(k+1)、xi3(k+1) represent that the i-th axle servo-drive system position exports x respectivelyi1(k), motor speed xi2(k), new expansion state amount xi3(k) in the value of+1 sampling instant of kth；Step 2) establishes networking multiaxial motion position synchronization coupling error model；Isochronous controller of the step 3) design based on linear active disturbance rejection control, realizes networking multiaxial motion position synchronization control.
- 2. a kind of networking multiaxial motion position synchronization control method based on automatic disturbance rejection controller as claimed in claim 1, Characterized in that, in the step 2), the process for establishing networking multiaxial motion position synchronization coupling error model is as follows：2.1) defining multiaxis position synchronous error model isε (k)=Γ e (k) (6)Wherein, ε (k), e (k) are respectively multiaxis position synchronous error vector sum multiaxis position error vector, and ε (k)=[ε1 (k),…,εi(k),…,εn(k)] ', e (k)=[e11(k),…,ei1(k),…,en1(k)] ', symbol " ' " representing matrix turns Put, εi(k)、ei1(k) the i-th (i=1 ..., n) shaft position synchronous error and site error are represented respectively, and Γ represents synchronous conversion square Battle array；The synchronous transformation matrix Γ of selection is as follows：I.e. position synchronous error represents as follows：<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&epsiv;</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <msub> <mi>e</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mn>21</mn> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>+</mo> <msub> <mi>e</mi> <mrow> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&epsiv;</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <msub> <mi>e</mi> <mn>21</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mn>31</mn> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>+</mo> <msub> <mi>e</mi> <mn>11</mn> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&epsiv;</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>+</mo> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&epsiv;</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <msub> <mi>e</mi> <mrow> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>e</mi> <mn>11</mn> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>+</mo> <msub> <mi>e</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>2.2) establishing multiaxis position synchronization coupling error model isE (k)=e (k)+α ε (k) (9)Wherein, E (k)=[E1(k),…,Ei(k),…,En(k)], and α is diagonal and positive definite control gain matrix, by formula (6) (9) are substituted into obtainE (k)=(I+ α Γ) e (k) (10)Wherein, I represents unit matrix, when (I+ α Γ) can the inverse time, E (k) → 0 can release e (k) → 0, further, by e (k) → 0 can release ε (k) → 0.
- A kind of 3. networking multiaxial motion position synchronization control side based on automatic disturbance rejection controller as claimed in claim 1 or 2 Method, it is characterised in that in the step 3), the design process for designing the isochronous controller based on linear active disturbance rejection control is as follows：3.1) to the i-th axle Servo Control System Design linearity tracking differentiator, i=1 ..., n, transition process arranging, Setting signal v0As reference input, position v is obtained by Nonlinear Tracking Differentiator0Approximate velocity differential signal vi2(k), while given Signal v0Transition value vi1(k), the reference signal of setting is smoothed, prevents excessive overshoot, its form is as follows：<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>fh</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msubsup> <mi>r</mi> <mrow> <mi>i</mi> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <msub> <mi>v</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>-</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mn>0</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>v</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Tv</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>v</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>Tfh</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>Wherein, ri0For the Turbo Factor of Nonlinear Tracking Differentiator, fhi(k) it is vi2Differential value, T is the sampling period；3.2) to the i-th axle Servo Control System Design linear extended state observer, state and summation disturbance to system are carried out Estimation in real time and compensation, its form are as follows：<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>e</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>-</mo> <msub> <mi>&beta;</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>e</mi> <mi>i</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mn>3</mn> </mrow> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>-</mo> <msub> <mi>&beta;</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>e</mi> <mi>i</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow> <mi>i</mi> <mn>0</mn> </mrow> </msub> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mn>3</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mn>3</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>T</mi> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>&beta;</mi> <mrow> <mi>i</mi> <mn>3</mn> </mrow> </msub> <msub> <mi>e</mi> <mi>i</mi> </msub> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>Wherein, ei(k) it is the i-th axle servo-drive system physical location and the difference of its estimate, i.e. the position margin of error, zi1(k) it is to i-th Axle servo-drive system position xi1(k) estimation, zi2(k) it is to speed xi2(k) estimation, zi3(k) it is i.e. total to new expansion state amount With disturbance xi3(k) estimation, βi1、βi2、βi3For one group of parameter to be adjusted, to ensure estimated accuracy, seen according to high gain state Survey device design principle, βi1、βi2、βi3Value is more than the upper bound of noise or disturbance, and takes β by POLE PLACEMENT USINGi1=3 ωi0, βi2= 3ωi0 2,ωi0For observer bandwidth, bi0For adjustable compensating factor；3.3) for the uncertain and summation disturbance as caused by time-vary delay system in compensation system, obtained in the i-th axle servo-drive system Controlled quentity controlled variable ui0(k) z is subtracted ini3(k) new controlled quentity controlled variable, i.e. u are obtainedi(k)=ui0(k)-zi3(k)/bi0, wrap in bucking-out system Disturbed containing summation caused by time-vary delay system, eliminate influence of the time-vary delay system to systematic function, design has disturbance compensation as follows Synchronization Control rule：<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>e</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>u</mi> <mrow> <mi>i</mi> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>&lsqb;</mo> <mn>0</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>i</mi> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>0</mn> <mo>&rsqb;</mo> </mrow> <mo>&prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mi>p</mi> </msub> <mi>E</mi> <mo>+</mo> <msub> <mi>K</mi> <mi>d</mi> </msub> <mover> <mi>E</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>K</mi> <mi>e</mi> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>I</mi> <mo>+</mo> <mi>&alpha;</mi> <mi>&Gamma;</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msup> <mrow> <mo>&lsqb;</mo> <mn>0</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>i</mi> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mn>0</mn> <mo>&rsqb;</mo> </mrow> <mo>&prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mi>p</mi> </msub> <mo>(</mo> <mrow> <mi>I</mi> <mo>+</mo> <mi>&alpha;</mi> <mi>&Gamma;</mi> </mrow> <mo>)</mo> <mi>e</mi> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>K</mi> <mi>D</mi> </msub> <mrow> <mo>(</mo> <mi>I</mi> <mo>+</mo> <mi>&alpha;</mi> <mi>&Gamma;</mi> <mo>)</mo> </mrow> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>K</mi> <mi>e</mi> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>I</mi> <mo>+</mo> <mi>&alpha;</mi> <mi>&Gamma;</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>(</mo> <mi>k</mi> <mo>)</mo> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>u</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mrow> <mi>i</mi> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mn>3</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>b</mi> <mrow> <mi>i</mi> <mn>0</mn> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>Wherein, Kp、KdAnd KeTo control gain matrix,ei1(k) it is the i-th axle The Setting signal transition value v of servo-drive systemi1(k) with position estimation value zi1(k) error, ei2(k) given for the i-th axle servo-drive system Determine the differential value v of signali2(k) with velocity estimation value vi2(k) error, ui0(k) it is synchronous error feedback control amount, ui(k) it is Final controlled quentity controlled variable.
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