CN107247411A - Non-singular terminal sliding formwork Track In Track control method based on disturbance observer - Google Patents
Non-singular terminal sliding formwork Track In Track control method based on disturbance observer Download PDFInfo
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- G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
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Abstract
The invention discloses a kind of non-singular terminal sliding formwork Track In Track control method based on disturbance observer, with following steps：Set up the above water craft equation of motion for representing current ship kinetic characteristic and expect to carry transposed matrix R (ψ) in ship course model, the above water craft equation of motion；By Coordinate Conversion, by the above water craft equation of motion with expect ship course model conversion into standard second order nonlinear control system；Analysis obtains the error system in second nonlinear control system；When external disturbance meets following condition：Wherein, n is positive integer, Pi=diag (pi,1,pi,2,pi,3), and pi,j(j=1,2,3) it is arithmetic number；Finite time flight tracking control rule and corresponding disturbance observer are provided, Track In Track control is completed.
Description
Technical field
The present invention relates to a kind of non-singular terminal sliding formwork Track In Track control method based on disturbance observer.It is related to patent
Classification number G05 is controlled；Adjust the general controls of G05B or regulating system；The functional unit of this system；For this system or
The monitoring of unit or test device G05B13/00 adaptive control systems, i.e. system are according to some predetermined criterion adjust automaticallies
Oneself with the system G05B13/02 electricity of optimum performance G05B13/04 including the use of model or simulator.
Background technology
In field of non-linear control, finite-time control method has obtained widely studied due to its fast convergence.
Conventional finite-time control algorithm includes adding exponential integral, terminal sliding mode etc..In addition, thering is scholar to prove, in system asymptotically stability
On the basis of, if being able to demonstrate that, its degree of homogeneity is less than zero, then closed-loop system can reach the control effect of finite time stability.
Traditional can not be to outside time-varying uncertain disturbance based on the minus finite-time control method of degree of homogeneity
Handled, when its exterior disturbance is larger, system robustness is poor, and control performance declines.The invention is limited by introducing
Time disturbance observer so that system can effectively recognize outside uncertain disturbance, and closed-loop system is when meeting global limited
Between stable control effect, improve the robustness of control system.
The content of the invention
The present invention is directed to the proposition of problem above, and a kind of non-singular terminal sliding formwork based on disturbance observer developed navigates
Mark tracking and controlling method, with following steps：
- set up the above water craft equation of motion for representing current ship kinetic characteristic and expect ship course model, the water
Transposed matrix R (ψ) is carried in the ship equation of motion of face；
- pass through Coordinate Conversion, by the above water craft equation of motion with expect ship course model conversion into standard second order
Nonlinear control system；
- analysis obtains the error system in second nonlinear control system；
- when external disturbance meets following condition：
Wherein, n is positive integer, Pi=diag (pi,1,pi,2,pi,3), and pi,j(j=1,2,3) it is arithmetic number；
Finite time flight tracking control rule and corresponding disturbance observer are provided, Track In Track control is completed；
Described flight tracking control rule is as follows：
In formula,Wherein,WithObtained by the observation of following disturbance observer；
The disturbance observer is as follows：
In formula：
Θ0=ζe, u=RM-1τ+χe(·)
Wherein, τ is derived by by flight tracking control rule formula, Qi=diag (Qi,1,Qi,2,Qi,3), i=0,1 ..., n-1 are just
Constant diagonal matrix,And meet
As preferred embodiment, the described current above water craft equation of motion：
In formula：η=[x, y, ψ]TPosition (x, y) and deflection (ψ) of the expression above water craft under terrestrial coordinate system, ν=
[u,v,r]TThe linear velocity (u, v) and angular speed (r) of ship are represented, M is ship quality, meets M=MT>0, C (ν) is Ke Liao
Sharp centripetal force matrix, D (ν) is damping matrix, τ=[τ1,τ2,τ3]TIt is control input, (η is t) that the system of lump is not known to d
, R (ψ) is a transposed matrix, is expressed as：
R (ψ) has following property：
Property 1：RT(ψ) R (ψ)=I；
Property 2：To arbitrary ψ, haveAnd RT(ψ) S (r) R (ψ)=R (ψ) S (r) RT(ψ)=S (r),
And：
As preferred embodiment, described ship desired course is as follows：
Wherein, ηd=[xd,yd,ψd]TAnd νd=[ud,vd,rd]TIt is to expect ship motion state.
Brief description of the drawings
, below will be to embodiment or existing for clearer explanation embodiments of the invention or the technical scheme of prior art
There is the accompanying drawing used required in technology description to do one simply to introduce, it should be apparent that, drawings in the following description are only
Some embodiments of the present invention, for those of ordinary skill in the art, on the premise of not paying creative work, may be used also
To obtain other accompanying drawings according to these accompanying drawings.
Fig. 1-6 is not consider the simulation analysis result schematic diagram of external disturbance in the embodiment of the present invention
Simulation analysis result schematic diagram when Fig. 7-12 is considers that external disturbance meets hypothesis 1 in the embodiment of the present invention
Simulation analysis result schematic diagram when Figure 13-19 is considers that external disturbance meets hypothesis 2 in the embodiment of the present invention
Embodiment
To make the purpose, technical scheme and advantage of embodiments of the invention clearer, with reference to the embodiment of the present invention
In accompanying drawing, clear complete description is carried out to the technical scheme in the embodiment of the present invention：
A kind of non-singular terminal sliding formwork Track In Track control method based on disturbance observer, mainly comprises the following steps：
Consider that the above water craft equation of motion is as follows first：
In formula：η=[x, y, ψ]TPosition (x, y) and deflection (ψ) of the expression above water craft under terrestrial coordinate system, ν=
[u,v,r]TThe linear velocity (u, v) and angular speed (r) of ship are represented, M is ship quality, meets M=MT>0, C (ν) is Ke Liao
Sharp centripetal force matrix, D (ν) is damping matrix, τ=[τ1,τ2,τ3]TIt is control input, (η is t) that the system of lump is not known to d
, R (ψ) is a transposed matrix, is expressed as
Also, R (ψ) has the following property：
Property 1：RT(ψ) R (ψ)=I；
Property 2：To arbitrary ψ, haveAnd RT(ψ) S (r) R (ψ)=R (ψ) S (r) RT(ψ)=S (r),
And
Consider that ship desired course is as follows：
Wherein, ηd=[xd,yd,ψd]TAnd νd=[ud,vd,rd]TIt is to expect ship motion state.
This paper control targe is one control law τ of design so that actual signal (1) can be tracked in finite time
Desired signal (3).
Global finite time Track In Track controller design based on disturbance observer
Consider following coordinate transform
ζ=R (ψ) ν (4a)
ζd=R (ψd)νd (4b)
Wherein, ζ=[ζ1,ζ2,ζ3]T, ζd=[ζd,1,ζd,2,ζd,3]T.Hereinafter, we will use R and RdTo represent respectively
R (ψ) and R (ψd)。
It can be obtained by (1) and (4a)
Wherein δ (t)=RM-1(η, is t) external disturbance to d, in formula：
χ (η, ζ)=[s (ζ3)-RM-1(C(RTζ)+D(RTζ))RT]ζ-RM-1G (η, RTζ) (6)
Similarly, it can be obtained by (3) and (4b)
Make ηe=η-ηd, ζe=ζ-ζd.It can be obtained by (5) and (7)
In formula：
χe()=χ ()-S (ζD, 3)ζd-RM-1f(·) (9)
Wherein, ηe=[ηE, 1, ηE, 2, ηE, 3]T, ζe=[ζE, 1, ζE, 2, ζE, 3]T。
For error system (8)-(9), we will design the global finite time flight path based on disturbance observer below
Tracking control unit so that error ηeAnd ζeIn Finite-time convergence to zero.
The nominal design of control law of non-singular terminal sliding formwork of external disturbance is not considered
Define non-singular terminal sliding formwork control ratio
Wherein, s=[s1, s2, s3]T, p and q are positive odd numbers, andβ is normal number.
By using feedback linearization method, design NTSM control laws are as follows：
In formula wherein, k is positive controller design parameter, sgn (s)=[sign (s1), sign (s2), sign (s3)]T,
And
In the case of external disturbance is not considered, under control law (11) effect, closed-loop system (8)-(9) and (11)-(13)
Being capable of finite time stability.
Consider following Lyapunov functions
(14) derivation can be obtained
Work as ζe,iWhen ≠ 0, because p and q are positive odd numbers, andObviouslyIt can thus be concluded that
Wherein,Therefore, closed-loop system can reach NTSM sliding-mode surfaces in finite time
(10)。
Work as ζe,iWhen=0, control law (11) is brought into (8)-(9), obtained
Namely
Obviously, s is worked asi>When 0,Work as si<When 0,Therefore,It is not a domain of attraction, system is missed
Difference can be in finite timeReach NTSM sliding-mode surfaces (10).
Work as si=0, it can be obtained by (10)
Namely
It can thus be concluded that ηe,iCan be along NTSM sliding-mode surfaces (10) in finite timeReach zero point.
Analysis can be obtained more than, closed-loop system (8)-(9) and (11)-(13) finite time stability.Theorem must be demonstrate,proved.Note 1：
As p=q=1, NTSM sliding-mode surfaces (10) and NTSM control laws (11) will be degenerated to
τσ=M R-1[-βζe-k sgn(σ)]-M R-1χe(·) (21b)
Inference 1:Under control law (21b) effect, tracking error ηeAnd ζeNTSM sliding-mode surfaces will be reached in finite time,
Then exponential convergence is to zero point.
Prove：It is defined as follows Lyapunov functions
Its derivation can be obtained
It can be obtained according to theorem 1, tracking error ηe,iAnd ζe,iSliding-mode surface can be reached in finite time.
Once error reaches sliding-mode surface, it can be obtained by (8) and (21b)
It can thus be concluded that, error system can converge to zero point with exponential form.Inference 1 must be demonstrate,proved.
NTSM control laws control effect is analyzed when having disturbance
Assuming that 1：External disturbance δ (t) boundeds, i.e.,Wherein
In the case where meeting the external disturbance for assuming 1 condition effect, if control law parameter k is previous more than disturbing, then tracking
Error ηeAnd ζeZero point can be reached in finite time.
Prove：It can be obtained by (8), (11) and (14)
ChooseAccording to theorem 1, can obtain system can reach NTSM sliding-mode surfaces, Ran Hou in finite time
Finite-time convergence is to zero point.Thus, theorem must be demonstrate,proved.
Note 1：In order to ensure closed-loop system stability, it is necessary to select larger k so that k be more than disturbance it is previous.So
And, for sliding formwork control, k increase will cause the shake of actuator, and then be had undesirable effect to controller.
Note 2：In actual control system, due to the uncertainty of disturbance, it is difficult to estimate a suitable perturbating upper bound, this
If sample k selection is improper, closed-loop system will not be able to stabilization.
There is FTDO-NTSM Trajectory Tracking Controls rule design during external disturbance
Assuming that 2：Assuming that external disturbance is met
Wherein, n is positive integer, Pi=diag (pi,1,pi,2,pi,3), and pi,j(j=1,2,3) it is arithmetic number.
Consider to meet the external disturbance δ (t) for assuming 2, using feedback linearization method, design FTDO-NTSM finite times
Flight tracking control rule is as follows
In formula
Wherein,WithObtained by the observation of following disturbance observer
In formula：
Θ0=ζe, u=RM-1τ+χe(·) (30)
Wherein, τ is derived by by (11), Qi=diag (Qi,1,Qi,2,Qi,3), i=0,1 ..., n-1 are that normal number is diagonal
Battle array,And meet
Tracking error η so in (11)eAnd ζeSliding-mode surface can be reached in finite time and converge to zero point.First
Prove ηeAnd ζeNTSM sliding-mode surfaces can be reached in finite time.
Choose following Lyapunov functions
Its derivation can be obtained
It can obtain in finite timeIt can thus be concluded that
It is obvious that working as ζe,iWhen ≠ 0, systematic error state can be in Finite-time convergence to NTSM sliding-mode surfaces.
Work as ζe,iWhen=0, τABring error system (8)-(9) into, can obtain
Namely
According to theorem 1, can obtain systematic error can reach sliding-mode surface in finite time, and reach zero along sliding-mode surface
Point.So far, theorem must be demonstrate,proved.
Note 3：Work as p=q=1, FTDO-NTSM control laws will be degenerated to traditional Exponential Stability control based on disturbance observer
Device (DO-ESC) processed, corresponding control law is as follows：
Wherein, σ is provided by (21a), and Γ () is defined by (12),It is derived by by (29).
The suppression of actuator shake
In order to eliminate the phenomenon of the NTSM and FTDO-NTSM control laws actuator shake based on terminal sliding mode, introduce as follows
Saturation function [2]：
Wherein, λ>0,
If NTSM and FTDO-NTSM control laws are chosen as follows
Wherein, Γ () is provided by formula (12), and s is defined by (10).So, systematic error ηeAnd ζeCan be in finite time
It is interior to be calmed to zero point.
Prove：From theorem 1, systematic error can be calmed in finite time and be arrived | si|≤λ, i=1,2,3.Cause
This, can be obtained by (15)
Therefore, ζ is worked ase,iWhen ≠ 0, Lyapunov stability criterias are met.
Work as ζe,iWhen=0, control law (38b), which is substituted into (8)-(9), to be obtained
Obviously, whenWhen,WhenWhen,Therefore,It is not a domain of attraction.It can thus be concluded that systematic error ηeAnd ζeIt can be calmed in finite time to zero point.Similarly may be used
Also ensure that the stability in finite time of closed-loop system.So far, theorem must be demonstrate,proved.
Embodiment
Simulation analysis during external disturbance are not considered
NTSM control law (11) design parameter is accordingly：K=3.5, β=1, p=5, q=3, λ=6.8,
Corresponding simulation result is as shown in figures 1 to 6.
As Figure 1-3, compared to traditional ESC control laws (21b), under the effect of NTSM control laws, actual flight path energy
It is enough that upper desired track is tracked with faster convergence rate.And tracking error can be in Finite-time convergence to zero, such as Fig. 4-5 institutes
Show.As seen from Figure 6, in the presence of saturation function (37), controller input jiffer is substantially suppressed.
Consider simulation analysis during external disturbance
Disturbance, which is met, assumes 1
Controller parameter is chosen as follows：K=3.5, β=1, p=5, q=3, λ=6.8,
External disturbance isCorresponding simulation result such as Fig. 7-12.Can be with from figure
Find out, whenWhen, the NTSM control laws (11) proposed still are able to ensure the stability of closed-loop system, and tracking effect is better than
Traditional ESC control laws (21b), and control input non-jitter.
Disturbance, which is met, assumes 2
Assuming that external disturbanceChoose following control law parameter：K=2.2, β=1, p
=5, q=3, λ=6.8,Q0=diag (6,6,6), Q1=diag (42,42,42), Q2=diag (48,48,48),
P1=0, P2=diag (- 0.03 π2,-0.04π2,-0.02π2), simulation result is as shown in Figure 13-19.
It can be seen that being controlled compared to traditional NTSM control laws (11), ESC control laws (21b) and DO-ESC
System rule (36) proposed FTDO-NTSM control laws (27) enable to actual signal to track expectation at faster speed
Signal, tracking error can converge to zero, and external disturbance can be estimated in finite time and obtain, moreover,
Control input non-jitter in the presence of saturation function.
The foregoing is only a preferred embodiment of the present invention, but protection scope of the present invention be not limited thereto,
Any one skilled in the art the invention discloses technical scope in, technique according to the invention scheme and its
Inventive concept is subject to equivalent substitution or change, should all be included within the scope of the present invention.
Claims (3)
1. a kind of non-singular terminal sliding formwork Track In Track control method based on disturbance observer, it is characterised in that with following step
Suddenly：
- set up the above water craft equation of motion for representing current ship kinetic characteristic and expect ship course model, the waterborne vessel
Transposed matrix R (ψ) is carried in the oceangoing ship equation of motion；
- pass through Coordinate Conversion, by the above water craft equation of motion with expect ship course model conversion into standard second order non-thread
Property control system；
- analysis obtains the error system in second nonlinear control system；
- when external disturbance meets following condition：
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Wherein, n is positive integer, Pi=diag (pi,1,pi,2,pi,3), and pi,j(j=1,2,3) it is arithmetic number；
Finite time flight tracking control rule and corresponding disturbance observer are provided, Track In Track control is completed；
Described flight tracking control rule is as follows：
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In formula,Wherein,WithObtained by the observation of following disturbance observer；
The disturbance observer is as follows：
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In formula：
Θ0=ζe, u=RM-1τ+χe(·)
Wherein, τ is derived by by flight tracking control rule formula, Qi=diag (Qi,1,Qi,2,Qi,3), i=0,1 ..., n-1 are normal number
Diagonal matrix,And meet
2. the non-singular terminal sliding formwork Track In Track control method according to claim 1 based on disturbance observer, it is special
Levy and also reside in：
The described current above water craft equation of motion：
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<mo>(</mo>
<mi>v</mi>
<mo>)</mo>
<mi>v</mi>
<mo>-</mo>
<mi>D</mi>
<mo>(</mo>
<mi>v</mi>
<mo>)</mo>
<mi>v</mi>
<mo>-</mo>
<mi>g</mi>
<mo>(</mo>
<mi>&eta;</mi>
<mo>,</mo>
<mi>v</mi>
<mo>)</mo>
<mo>+</mo>
<mi>d</mi>
<mo>(</mo>
<mi>&eta;</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mtd>
</mtr>
</mtable>
</mfenced>
In formula：η=[x, y, ψ]TPosition (x, y) and deflection (ψ) of the expression above water craft under terrestrial coordinate system, ν=[u, v,
r]TThe linear velocity (u, v) and angular speed (r) of ship are represented, M is ship quality, meets M=MT>0, C (ν) is that Coriolis is centripetal
Torque battle array, D (ν) is damping matrix, τ=[τ1,τ2,τ3]TControl input, d (η, t) be lump system indeterminate, R (ψ)
It is a transposed matrix, is expressed as：
<mrow>
<mi>R</mi>
<mrow>
<mo>(</mo>
<mi>&psi;</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mi>&psi;</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<mi>&psi;</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>s</mi>
<mi>i</mi>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mi>&psi;</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>cos</mi>
<mrow>
<mo>(</mo>
<mi>&psi;</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
R (ψ) has following property：
Property 1：RT(ψ) R (ψ)=I；
Property 2：To arbitrary ψ, haveAnd RT(ψ) S (r) R (ψ)=R (ψ) S (r) RT(ψ)=S (r), and：
<mrow>
<mi>S</mi>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<mi>r</mi>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>r</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>.</mo>
</mrow>
3. the non-singular terminal sliding formwork Track In Track control method according to claim 1 based on disturbance observer, it is special
Levy and also reside in：
Ship desired course is as follows：
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<msub>
<mover>
<mi>&eta;</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>d</mi>
</msub>
<mo>=</mo>
<mi>R</mi>
<mo>(</mo>
<msub>
<mi>&psi;</mi>
<mi>d</mi>
</msub>
<mo>)</mo>
<msub>
<mi>v</mi>
<mi>d</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>M</mi>
<msub>
<mover>
<mi>v</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>d</mi>
</msub>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&eta;</mi>
<mi>d</mi>
</msub>
<mo>,</mo>
<msub>
<mi>v</mi>
<mi>d</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
Wherein, ηd=[xd,yd,ψd]TAnd νd=[ud,vd,rd]TIt is to expect ship motion state.
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| CN109460043A (en) * | 2018-12-29 | 2019-03-12 | 上海海事大学 | One kind being based on multi-modal non-singular terminal sliding formwork ship track Auto-disturbance-rejection Control |
| CN110247592A (en) * | 2019-06-28 | 2019-09-17 | 西安理工大学 | Introduce the multivariable second order non-singular terminal sliding formwork current control method of Reaching Law |
| CN110377044A (en) * | 2019-07-24 | 2019-10-25 | 东南大学 | A kind of the finite time height and Attitude tracking control method of unmanned helicopter |
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| CN108416082A (en) * | 2018-01-19 | 2018-08-17 | 中国人民解放军92859部队 | A kind of marine site pour point external disturbance gravitation horizontal component without unusual computational methods |
| CN108416082B (en) * | 2018-01-19 | 2021-09-10 | 中国人民解放军92859部队 | Singularity-free calculation method for external disturbance gravity horizontal component of sea area flow point |
| CN108681246A (en) * | 2018-05-14 | 2018-10-19 | 大连海事大学 | A kind of unmanned water surface ship Track In Track method based on actuator failures |
| CN108828955A (en) * | 2018-08-16 | 2018-11-16 | 大连海事大学 | Accurate Track In Track control method based on finite time extended state observer |
| CN109460043A (en) * | 2018-12-29 | 2019-03-12 | 上海海事大学 | One kind being based on multi-modal non-singular terminal sliding formwork ship track Auto-disturbance-rejection Control |
| CN109460043B (en) * | 2018-12-29 | 2022-08-19 | 上海海事大学 | Multimode nonsingular terminal sliding mode based ship track active disturbance rejection control method |
| CN110247592A (en) * | 2019-06-28 | 2019-09-17 | 西安理工大学 | Introduce the multivariable second order non-singular terminal sliding formwork current control method of Reaching Law |
| CN110377044A (en) * | 2019-07-24 | 2019-10-25 | 东南大学 | A kind of the finite time height and Attitude tracking control method of unmanned helicopter |
| CN113110532A (en) * | 2021-05-08 | 2021-07-13 | 哈尔滨工程大学 | Benthonic AUV self-adaptive terminal sliding mode trajectory tracking control method based on auxiliary dynamic system |
| CN113110532B (en) * | 2021-05-08 | 2022-11-04 | 哈尔滨工程大学 | Benthonic AUV self-adaptive terminal sliding mode trajectory tracking control method based on auxiliary dynamic system |
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