CN109212970A - A kind of neural network dynamic face control method of drive lacking rope system complex system - Google Patents

A kind of neural network dynamic face control method of drive lacking rope system complex system Download PDF

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CN109212970A
CN109212970A CN201811088097.0A CN201811088097A CN109212970A CN 109212970 A CN109212970 A CN 109212970A CN 201811088097 A CN201811088097 A CN 201811088097A CN 109212970 A CN109212970 A CN 109212970A
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CN109212970B (en
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黄攀峰
鲁迎波
孟中杰
张帆
张夷斋
刘正雄
董刚奇
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Northwestern Polytechnical University
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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 neural network dynamic face control methods of drive lacking rope system complex system, comprising the following steps: will arrest rear space rope system complex system and be decomposed into drive lacking subsystem ΞbWith full drives subsystem Ξa, and establish rope system drive lacking subsystem ΞbWith full drives subsystem ΞaKinetic model, drive lacking subsystem ΞbWith full drives subsystem ΞaBetween there are coupled relations: for exist disturbance drive lacking subsystem ΞbControl, using neural network and PD control design virtual controlling rule, construct desired augmentation pursuit path；For the full drives subsystem Ξ that there is disturbance and control input-bounda, using desired augmentation pursuit path as tracking amount, design neural network dynamic face control law.This method can be used for solving inputting the stable control that rope under saturated conditions is complex there are external disturbance and control；It can be used for solving the control problem of drive lacking rope system system.

Description

A kind of neural network dynamic face control method of drive lacking rope system complex system
Technical field
The present invention relates to the neural network dynamic face control method that a kind of target arrests rear drive lacking rope system complex system, Belong to the scope that rope is spacecraft stability contorting.
Background technique
Rope system spacecraft is by the tether of light soft (to fly tongue, fly lance, winged space platform and spacecraft, catching device Net etc.) connect and the complex system that is formed.Rope system spacecraft can be used for completing track rubbish (such as space junk, rocket Upper Stage etc.) it removes, the space tasks such as maintainable technology on-orbit of failure spacecraft.Above-mentioned task is needed in complex system attitude stabilization Under the premise of execute.After forming complex after catching device captures high speed rotation unstability noncooperative target, to realize complex Posture it is calm, provide desired control moment using the tenslator on space platform, cooperate on catching device Control mechanism, complete to arrest the stability contorting of rear complex, be convenient for subsequent task-cycle.
For special rope system spacecraft, tether length only has even hundreds of meters, far smaller than conventional cord system of tens of rice The tether length of satellite, it is therefore desirable to consider to arrest the 3 d pose of rear complex, classical " dumbbell body " mould in tethered satellite Type is no longer applicable in.Rope system system is typical drive lacking, a close coupling, nonlinearity system, in addition the shadow of external disturbance It rings, control becomes complex.Between in the past few decades, the research for the control of drive lacking rope system's system is had been achieved for Certain progress., Harbin Institute of Technology's grandson's radiance etc., Canadian York University it is concentrated mainly on following aspects: 1) Zhu Zhenghong etc. devises fractional order control rule, realizes the control of tethered satellite " dumbbell body " drive lacking model；2), Nanjing aviation is navigated Its university text is great etc. to have studied the stability contorting of tethered satellite during electric power tether recycling/release, devises corresponding system Rope tension control force and current control rule, but also only considered three the face interior angle of tether, face exterior angle and tether length variables, Have ignored the posture of satellite；3), Northwestern Polytechnical University Huang Panfeng teach team have studied rope system robot target arrest it is rear compound The stability contorting of body, it is contemplated that arrest the 3 d pose of rear complex, but it is full drive system that its hypothesis, which arrests rear complex system,.
Therefore up to the present, consider that the drive lacking rope system systematic research of target 3 d pose is relatively fewer, hold in addition Row device control input-bound, rope are the limitation of the constraints such as exterior disturbance, for such drive lacking rope system complex system Control has actual engineering significance and researching value.
Summary of the invention
In order to solve the problems existing in the prior art, the invention proposes a kind of nerve nets of drive lacking rope system complex system Network dynamic surface control method is asked with processing space rope system complex system by external disturbance, control input-bound and drive lacking Topic, for there is the full drives subsystem Ξ a for disturbing and controlling input-bound, using desired augmentation pursuit path as tracking amount, if Neural network dynamic face control law is counted, realizes that system is stablized.
To achieve the above object, the present invention is to be achieved through the following technical solutions:
A kind of neural network dynamic face control method of drive lacking rope system complex system, comprising the following steps:
Rear space rope system complex system will be arrested and be decomposed into drive lacking subsystem ΞbWith full drives subsystem Ξa, and establish Rope system drive lacking subsystem ΞbWith full drives subsystem ΞaKinetic model, drive lacking subsystem ΞbWith full drives subsystem Ξa Between there are coupled relations:
For the drive lacking subsystem Ξ that there is disturbancebControl, virtual controlling is designed using neural network and PD control Rule, constructs desired augmentation pursuit path；
For the full drives subsystem Ξ that there is disturbance and control input-bounda, it is tracking with desired augmentation pursuit path Amount designs neural network dynamic face control law.
As a further improvement of the present invention, shown in kinetics equation such as formula (1)~(6) of space rope system complex system:
Wherein,θ2,ψ2, α, β, l represents system generalized coordinates,Qα,Qβ,QlRepresent generalized force or torque；With() is represented to the one of the time, second-order differential, m1、m2It represents space platform, arrest rear composite quality, both definition Gross mass be m=m1+m2, b2yFor the distance of tether tie point to target centroid, Ix,Iy,IzFor three axis for arresting rear complex Rotary inertia, ω0For orbit angular velocity, a4,a5,b4,b5,c1,c2,d1,d2,e1,e2,e3,f1,f2,g1,g2,A,λ11,λ12,λ13, λ14,λ15,λ16,λ17,λ18,λ19,χ1,χ2,χ3,χ4,χ5,χ6,χ7Detailed expressions it is as follows:
As a further improvement of the present invention, ΞbExpression formula is as follows:
Wherein:
ΞaExpression formula is as follows:
Wherein:
Its detailed expressions is as follows:
Formula (10) are substituted into formula (11), arrangement can obtain:
As a further improvement of the present invention, PD virtual controlling is restrained are as follows:
Define virtual control instructionπd=πd1+πd2, design πdPD virtual controlling rule are as follows:
πd1=b-1fπ (18)
Wherein,kb1,kb2For normal number to be designed, ξbdFor desired control System instruction.
As a further improvement of the present invention, the control law of neural network are as follows:
DefinitionNeural network weight estimated matrixMeet following adaptive law:
Wherein,For real number matrix to be designed, σ1For real constant；0, I is respectively 2 × 2 rank zero moments Battle array and unit matrix.
As a further improvement of the present invention, the Ξ of drive lacking subsystembSteps are as follows for design of control law:
Define ξb1=ξb=[α, β]T,db=[db1,db2]T,
Arrangement formula (10) are as follows:
Define virtual control instructionπd=πd1+πd2, design πdPD virtual controlling rule are as follows:
πd1=b-1fπ (18)
Wherein,kb1,kb2For normal number to be designed, ξbdFor desired control System instruction；DefinitionIt is approached using the universal approximation property of neural networkHave
Wherein, WπiFor neural network weight, SπiIt (z) is radial basis function, επ1,επ2Deviation is approached for neural network；It is fixed Adopted πd2Are as follows:
Wherein,ForEstimated value；DefinitionNeural network weight estimated matrixMeet as follows certainly Adapt to rule:
Wherein,For real number matrix to be designed, σ1For real constant；0, I is respectively 2 × 2 rank zero moments Battle array and unit matrix；So far to πdIntegral can obtain virtual expectation instruction∫ ∫ () dt is to the time Double integral；DefinitionFor desired augmentation pursuit path, ψd,ldFor preset desired value.
As a further improvement of the present invention, neural network dynamic face control law formula are as follows:
Wherein, K2∈R4×1For real number matrix to be designed；W3For neural network weight, Θ3(Z3) it is radial basis function, if It is fixedFor W3Estimated value, estimated biasForK2∈R4×1For real number matrix to be designed；W4For nerve net Network weight, Θ4(Z4) it is radial basis function, settingFor W4Estimated value, estimated biasFor.
As a further improvement of the present invention, full drives subsystem ΞaSteps are as follows for control design case:
DefinitionFormula (15) can transform to:
Define first dynamic surface state variable error are as follows:
z1=ξa1-ξad (23)
Design virtual controlling amount
Wherein, χ is positive real number；It is τ that α, which is input to time constant,2Low-pass filter, obtain new state variableIt is full Foot:
Define second dynamic surface state variable error are as follows:
To formula (26) derivation and bring into formula (22) arrange can obtain:
Using the model Uncertainty in the universal approximation property approximation of neural network compensation (15)
Wherein, W3For neural network weight, Θ3(Z3) it is radial basis function,ε3Have for minimum Boundary's modeling error；SettingFor W3Estimated value, estimated biasForEstimated value is online by following adaptive laws It updates:
Wherein,σ3For real constant；
Design control law τ accordinglycExpression formula are as follows:
Wherein, K2∈R4×1For real number matrix to be designed；In view of the saturated characteristic of control law τ, definition saturation vector Δ τ=τc- τ, saturation function meet such as minor function:
Wherein, τmax,τminFor the bound of control input, control input saturation problem still is solved with neural network；It is fixed Justice:
Wherein, W4For neural network weight, Θ4(Z4) it is radial basis function,ε4It is minimum Bounded modeling error；SettingFor W4Estimated value, estimated biasForEstimated value is by following adaptive laws Online updating:
Wherein,σ4For real constant；Accordingly, design control law formula are as follows:
Compared with prior art, the invention has the following advantages that
The kinetic model of space rope system complex system is decomposed into drive lacking channel and full driving channel, benefit by the present invention With the coupled characteristic of two interchannels, the stability contorting of 6 quantity of states is realized with 4 control amounts；And neural network control is designed with this System rule, online compensation external disturbance and control input saturation item, realize that rope is the stability contorting of complex.By for design Control law, which carries out Liapunov stability to system, proves that obtain entire closed-loop system asymptotically stability of the invention, state is inclined Difference, auxiliary variable and estimated bias are ultimately uniform boundaries.Therefore, the stability of control system must be demonstrate,proved.Therefore the present invention It can be used for solving inputting the stable control that rope under saturated conditions is complex there are external disturbance and control；It can be used for solving The control problem of drive lacking rope system system.
Detailed description of the invention
Fig. 1 present invention is that rope is system coordinate system definition figure after arresting；
Control system block diagram Fig. 2 of the invention.
Specific embodiment
It elaborates with reference to the accompanying drawing with implementation to the present invention.
The present invention is the problem of processing space rope system complex system is by external disturbance, control input-bound and drive lacking, The present invention provides a kind of neural network dynamic face control methods of drive lacking rope system complex system, are that will arrest rear complex System decomposition is drive lacking subsystem ΞbWith full drives subsystem Ξa, there are coupled relations between two subsystems；It is disturbed for existing Drive lacking subsystem ΞbControl, using neural network and PD control design virtual controlling rule, construct desired augmentation with Track track；For the full drives subsystem Ξ that there is disturbance and control input-bounda, it is tracking with desired augmentation pursuit path Amount designs neural network dynamic face control law, realizes that system is stablized.Specific step is as follows:
Step 1, establishing rope is drive lacking subsystem ΞbWith full drives subsystem ΞaKinetic model；
Step 2, drive lacking subsystem Ξ is designedbWith full drives subsystem ΞaControl law；
Step 3, carrying out Liapunov stability to system for the control law of design proves.
1, establishing rope is drive lacking subsystem ΞbWith full drives subsystem ΞaKinetic model
Fig. 1 is that rope is system coordinate system definition figure after arresting.Wherein, OXYZ is inertial coodinate system, and coordinate origin is located at ground At the heart；o0x0y0z0For system track coordinate system, origin o0In entire rope system centroid position, o0y0Along space rope system system Move tangential direction, o0x0Away from the earth's core；α is tether face interior angle, and β is tether face exterior angle, and l is tether length.
For simplicity, model is handled as follows: 1), regards tether as single hop massless rigid rod；2), visual space platform For particle, three attitude angles for arresting rear complex are consideredθ2,ψ2；3), entirely rope system operates in Kepler's circular orbit On；4), generalized force is zero outside in the face of tether and face；5) quality for, arresting rear complex is much larger than the quality of catching device.
It is derived according to Lagrangian method, obtains kinetics equation such as formula (1)~(6) institute of space rope system complex system Show:
Wherein,θ2,ψ2, α, β, l represents system generalized coordinates,Qα,Qβ,QlRepresent generalized force or torque；With() is represented to the one of the time, second-order differential, m1、m2It represents space platform, arrest rear composite quality, both definition Gross mass be m=m1+m2, b2yFor the distance of tether tie point to target centroid, Ix,Iy,IzFor three axis for arresting rear complex Rotary inertia, ω0For orbit angular velocity, a4,a5,b4,b5,c1,c2,d1,d2,e1,e2,e3,f1,f2,g1,g2,A,λ11,λ12,λ13, λ14,λ15,λ16,λ17,λ18,λ19,χ1,χ2,χ3,χ4,χ5,χ6,χ7Detailed expressions it is as follows:
Due to Qα,QβIt is zero, regards modeling error and external disturbance as the overall disturbance d=[d of boundeda1,da2,da3,db1, db2,da4]T, breakdown (1)~(6) are drive lacking subsystem ΞbWith full drives subsystem Ξa。
ΞbExpression formula is as follows:
Wherein:
ΞaExpression formula is as follows:
Wherein:
Its detailed expressions is as follows:
Formula (10) are substituted into formula (11), arrangement can obtain:
So far, establishing rope is drive lacking subsystem ΞbWith full drives subsystem ΞaKinetic model.
2, drive lacking subsystem ΞbWith with full drives subsystem ΞaDesign of control law
Complete control system block diagram is as shown in Fig. 2.Drive lacking subsystem ΞbWith full drives subsystem ΞaBetween exist coupling Relationship.Drive lacking subsystem ΞbControl law be made of PD virtual controlling rule and neural network virtual controlling rule, neural network is dynamic State face control law is for realizing full drives subsystem ΞaStability contorting.
(1) Ξ of drive lacking subsystembDesign of control law
Define ξb1=ξb=[α, β]T,db=[db1,db2]T,
Arrangement formula (10) are as follows:
Define virtual control instructionπd=πd1+πd2, design πdPD virtual controlling rule are as follows:
πd1=b-1fπ (18)
Wherein,kb1,kb2For normal number to be designed, ξbdFor desired control System instruction.DefinitionIt is approached using the universal approximation property of neural networkHave
Wherein, WπiFor neural network weight, SπIt (z) is radial basis function, επDeviation is approached for neural network.Define πd2 Are as follows:
Wherein,ForEstimated value, definitionNeural network weight estimated matrixMeet following adaptive Ying Lv:
Wherein,For real number matrix to be designed, σ1For real constant.So far to πdIntegral can obtain void Quasi- expectation instruction∫ ∫ () dt is the double integral to the time.DefinitionFor Desired augmentation pursuit path, ψd,ldFor preset desired value.
(2) full drives subsystem ΞaControl
Define ξa=ξa1,Formula (15) can transform to:
Define first dynamic surface state variable error are as follows:
z1=ξa1-ξad (23)
Design virtual controlling amount
Wherein, χ is positive real number.It is τ that α, which is input to time constant,2Low-pass filter, obtain new state variableIt is full Foot:
Define second dynamic surface state variable error are as follows:
To formula (26) derivation and bring into formula (22) arrange can obtain:
Using the model Uncertainty in the universal approximation property approximation of neural network compensation (15)
Wherein, W3For neural network weight, Θ3(Z3) it is radial basis function,ε3Have for minimum Boundary's modeling error.SettingFor W3Estimated value, estimated biasForEstimated value is online by following adaptive laws It updates:
Wherein,σ3For real constant.
Design control law τ accordinglycExpression formula are as follows:
Wherein, K2∈R4×1For real number matrix to be designed.In view of the saturated characteristic of control law τ, definition saturation vector Δ τ=τc- τ, saturation function meet such as minor function:
Wherein, τmax,τminFor the bound of control input.Still control input saturation problem is solved with neural network.It is fixed Justice:
Wherein, W4For neural network weight, Θ4(Z4) it is radial basis function,ε4It is minimum Bounded modeling error.SettingFor W4Estimated value, estimated biasForEstimated value is existed by following adaptive laws Line updates:
Wherein,σ4For real constant.Accordingly, design control law formula are as follows:
3, carrying out Liapunov stability to system for the controller of design proves
(1) drive lacking subsystem ΞbStability proves
The virtual controlling of design is restrained into πdSubstitution formula (17) and formula (18) arrange to obtain its error dynamics equation are as follows:
Wherein,0, I is respectively 2 × 2 rank null matrix and unit square Battle array.
To prove under-actuated systems ΞbStability, design following candidate liapunov function expression formula are as follows:
Calculate liapunov function V1Differential can obtain:
Wherein, | | | | the norm for being, and SπBounded meets | | Sπ||≤s*, Be positive reality Number.AbrFor Hurwitz matrix, there are Pbr, meet following relationship:QbrAll Eigenvalues be Just.
Calculate liapunov function V2Differential can obtain:
According to formula (37) and formula (38), can obtain:
Wherein:
Formula (39) two sides are integrated, can be obtained:
Wherein, VbIt (0) is VbInitial value, by formula (40) and liapunov function VbDefinition it is found that entire closed loop system System asymptotically stability, state deviationEstimated biasIt is ultimately uniform boundary.
(2) full drives subsystem ΞaStability proves
The following candidate liapunov function of design carries out the stability analysis of closed-loop system:
Wherein,Calculate VaTime diffusion, can obtain:
Due toFor antisymmetric matrix, for arbitrary vector x ∈ R4×1, perseverance hasIt is right Formula V1aCarry out derivation and substitute into formula (22) to arrange and can obtain:
Substitution formula (32) and control law formula (34), arranging formula (43) can obtain:
To Formula V2aDerivation, and substitute into adaptive law formula (29) and arrange and can obtain:
Known according to Young geometric theorem:
Therefore formula (45) can arrange are as follows:
To Formula V3aDerivation, and formula (24) and formula (25) are substituted into, it obtains
Wherein, continuous functionMeet | | B | |≤BM.DefinitionTherefore formula (48) Meet
Formula (44), (47) and (49) are substituted into formula (42) to arrange and can obtain:
Wherein:
Formula (50) two sides are integrated, can be obtained:
Wherein, VaIt (0) is VaInitial value, by formula (51) and liapunov function VaDefinition it is found that entire closed loop system System asymptotically stability, state deviation z2, auxiliary variable υ and estimated biasIt is ultimately uniform boundary.
Therefore, the stability of control system must be demonstrate,proved.
Although specific embodiments of the present invention are described in conjunction with attached drawing above, the invention is not limited to upper The specific embodiment stated, above-mentioned specific embodiment are only schematical, directiveness rather than restrictive.This The those of ordinary skill in field under the enlightenment of this specification, in the feelings for not departing from scope of the claimed protection of the invention Under condition, a variety of forms can also be made, these belong to the column of protection of the invention.

Claims (8)

1. a kind of neural network dynamic face control method of drive lacking rope system complex system, which is characterized in that including following step It is rapid:
Rear space rope system complex system will be arrested and be decomposed into drive lacking subsystem ΞbWith full drives subsystem Ξa, and establish rope system Drive lacking subsystem ΞbWith full drives subsystem ΞaKinetic model, drive lacking subsystem ΞbWith full drives subsystem ΞaBetween deposit In coupled relation:
For the drive lacking subsystem Ξ that there is disturbancebControl, using neural network and PD control design virtual controlling rule, construction Desired augmentation pursuit path out；
For the full drives subsystem Ξ that there is disturbance and control input-bounda, using desired augmentation pursuit path as tracking amount, if Count neural network dynamic face control law.
2. a kind of neural network dynamic face control method of drive lacking rope system according to claim 1 complex system, It is characterized in that, shown in kinetics equation such as formula (1)~(6) of space rope system complex system:
Wherein,System generalized coordinates is represented,Represent generalized force or torque；With () is represented to the one of the time, second-order differential, m1、m2It represents space platform, arrest rear composite quality, define total matter of the two Amount is m=m1+m2, b2yFor the distance of tether tie point to target centroid, Ix,Iy,IzIt is used to arrest the three axis rotation of rear complex Amount, ω0For orbit angular velocity, a4,a5,b4,b5,c1,c2,d1,d2,e1,e2,e3,f1,f2,g1,g2,A,λ11,λ12,λ13,λ14,λ15, λ16,λ17,λ18,λ19,χ1,χ2,χ3,χ4,χ5,χ6,χ7Detailed expressions it is as follows:
3. a kind of neural network dynamic face control method of drive lacking rope system according to claim 1 complex system, It is characterized in that,
ΞbExpression formula is as follows:
Ξb:
Wherein:
ΞaExpression formula is as follows:
Wherein:
Its detailed expressions is as follows:
Formula (10) are substituted into formula (11), arrangement can obtain:
4. a kind of neural network dynamic face control method of drive lacking rope system according to claim 3 complex system, It is characterized in that, PD virtual controlling rule are as follows:
Define virtual control instructionπd=πd1+πd2, design πdPD virtual controlling rule are as follows:
πd1=b-1fπ (18)
Wherein,kb1,kb2For normal number to be designed, ξbdRefer to for desired control It enables.
5. a kind of neural network dynamic face control method of drive lacking rope system according to claim 3 complex system, It is characterized in that, neural network virtual controlling rule are as follows:
DefinitionNeural network weight estimated matrixMeet following adaptive law:
Wherein,For real number matrix to be designed, σ1For real constant；0, I be respectively 2 × 2 rank null matrix and Unit matrix.
6. a kind of neural network dynamic face control method of drive lacking rope system according to claim 4 or 5 complex system, It is characterized in that,
The Ξ of drive lacking subsystembSteps are as follows for design of control law:
Definition
Arrangement formula (10) are as follows:
Define virtual control instructionπd=πd1+πd2, design πdPD virtual controlling rule are as follows:
πd1=b-1fπ (18)
Wherein,kb1,kb2For normal number to be designed, ξbdRefer to for desired control It enables；DefinitionIt is approached using the universal approximation property of neural networkHave
Wherein, WπiFor neural network weight, SπiIt (z) is radial basis function, επ1,επ2Deviation is approached for neural network；Define πd2 Are as follows:
Wherein,ForEstimated value；DefinitionNeural network weight estimated matrixMeet following adaptive Rule:
Wherein,For real number matrix to be designed, σ1For real constant；0, I be respectively 2 × 2 rank null matrix and Unit matrix；So far to πdIntegral can obtain virtual expectation instruction∫ ∫ () dt is to the two of the time Multiple integral；DefinitionFor desired augmentation pursuit path, ψd,ldFor preset desired value.
7. a kind of neural network dynamic face control method of drive lacking rope system according to claim 3 complex system, It is characterized in that, neural network dynamic face control law formula are as follows:
Wherein, K2∈R4×1For real number matrix to be designed；W3For neural network weight, Θ3(Z3) it is radial basis function, setting For W3Estimated value, estimated biasForK2∈R4×1For real number matrix to be designed；W4For neural network power Weight, Θ4(Z4) it is radial basis function, settingFor W4Estimated value, estimated biasFor.
8. a kind of neural network dynamic face control method of drive lacking rope system according to claim 7 complex system, It is characterized in that,
Full drives subsystem ΞaSteps are as follows for control design case:
DefinitionFormula (15) can transform to:
Define first dynamic surface state variable error are as follows:
z1=ξa1-ξad (23)
Design virtual controlling amount
Wherein, χ is positive real number；It is τ that α, which is input to time constant,2Low-pass filter, obtain new state variableMeet:
Define second dynamic surface state variable error are as follows:
To formula (26) derivation and bring into formula (22) arrange can obtain:
Using the model Uncertainty in the universal approximation property approximation of neural network compensation (15)
Wherein, W3For neural network weight, Θ3(Z3) it is radial basis function,ε3It is modeled for minimum bounded Error；SettingFor W3Estimated value, estimated biasForEstimated value is by following adaptive law online updatings:
Wherein,σ3For real constant；
Design control law τ accordinglycExpression formula are as follows:
Wherein, K2∈R4×1For real number matrix to be designed；In view of the saturated characteristic of control law τ, definition saturation vector Δ τ= τc- τ, saturation function meet such as minor function:
Wherein, τmax,τminFor the bound of control input, control input saturation problem still is solved with neural network；Definition:
Wherein, W4For neural network weight, Θ4(Z4) it is radial basis function,ε4For minimum bounded Modeling error；SettingFor W4Estimated value, estimated biasForEstimated value by following adaptive laws online more It is new:
Wherein,σ4For real constant；Accordingly, design control law formula are as follows:
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