CN107272417A - A kind of Neural Network Based Nonlinear control method of imitative operant conditioned reflex - Google Patents
A kind of Neural Network Based Nonlinear control method of imitative operant conditioned reflex Download PDFInfo
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Abstract
The invention discloses a kind of Neural Network Based Nonlinear control method of imitative operant conditioned reflex, including：Step 1: setting up operant conditioned reflex bionic model；Step 2: setting up the class Non-Affine Systems step 3 based on operant conditioned reflex bionic model：Utilize OCBM biometrics design controller u；Step 4：Controller u is applied to as above Non-Affine Systems, makes output y (t) by given accuracy β0Track desired trajectory xd(t), while ensuring system tracking error e (t) in the bounded of t >=0.Biological Principles of the present invention from operant conditioned reflex, construct a kind of artificial neural network by bionical inspiration and use it for handling the unknown nonlinear of complex uncertainty system, and the control strategy based on operant conditioned reflex model is designed for the uncertain Non-Affine Systems of a class, improve validity, flexibility and the adaptivity of ANN Control.
Description
Technical field
It is more particularly to a kind of to be used to handle multiple the present invention relates to Control of Nonlinear Systems field and ANN Control field
The method of the unknown nonlinear of miscellaneous uncertain system.
Background technology
Nonlinear function this feature that artificial neural network (ANN) can be approached in any norm by it, frequently as one
The mathematical tool gained great popularity is planted in the extensive use of Control of Nonlinear Systems field.In many achievements, nerve network controller
Three types can be divided into：The I type controllers of off-line training weights, the online weights learning II types control based on fixed network structure
Device, and the adjustable online weights learning III type controllers of network structure.
So far, most research work are concentrated in the design and development based on I types and II type controllers.So
And I type controllers lack the adaptive ability of offline Weight Training network, II type controllers require preset a large amount of in initialization
Neuron, causes control performance to depend greatly on the neuron number and corresponding structural parameters manually chosen, because
This phenomenon that would generally be consumed huge system operations resource and produce over-fitting.To break through the use limitation of II type controllers
Property, occurred in succession based on weights and the type III controller of network structure energy adjust automatically.Using automatically determine number of network node as
Design object, type III controller is intended to avoid passing through the computational burden that manual type introduced multi-neuron and caused and excessively
Parameterization.But the time that type III controller occurs is shorter, and its theoretical system is still unsound, and there is number of values must go deep into
The problem of discussing and be unpredictable.For example, the neutral net of suitable number neuron how is automatically generated during system operation,
How simplify control device structure and cumbersome stability analysis process, how to avoid that the parameter Estimation of excessive calculation resources can be consumed
How algorithm, optimize validity, flexibility and the adaptivity of network topology structure further to lift ANN Control etc.
Deng.These are required for using type III neuron controller as blank, the neural network control method with self-structuring ability are carried out excellent
Change with improving.
The content of the invention
In view of this, it is an object of the invention to provide a kind of Neural Network Based Nonlinear controlling party of imitative operant conditioned reflex
Method, with realize reduced in Control of Nonlinear Systems it is artificial adjust ginseng work, independent of accurate drift nonlinear transformations, without
Off-line training with more wide in range system operation conditions, and can ensure control accuracy with shutting down reprogram process
Consume less system operations resource simultaneously.
The Neural Network Based Nonlinear control method of the imitative operant conditioned reflex of the present invention, comprises the following steps：
Step 1: setting up operant conditioned reflex bionic model：
Neuron in network is classified by neururgic difference, constituted with identical neururgic neuron
One neural adaptive unit, universe network is made up of M neural adaptive unit, then the god of i-th of neural adaptive unit
Through activity：
Wherein, μi∈Rq, σi∈ R are the parameter of i-th of Neural cluster, z=[z1,z2,...,zq]TIt is defeated for neutral net
Enter；Nervous activity principle, μ are maximized according to reward behavioriObtained by following formula：
Wherein, riAt the time of representing i-th of bonus event generation.
Step 2: setting up the class Non-Affine Systems based on operant conditioned reflex bionic model：
Wherein, system mode vector x=[x1,...,xn]T∈Rn, system input/controller signals u ∈ R, system output
For y ∈ R；Fξ(t)() represents unknown nonlinear model of the structure with time drift, and concrete form is as follows：
Fξ(t)()={ Fi() | i=ξ (t) ∈ N+,0≤ti≤t<ti+1}
And Fξ(t)() meetsThere is bounded positive definite scalar function a (x), b (x), c (x) so that
|Fξ(t)(x, 0) |≤c (x),Set up；
Define system mode error vector e (t) and filtering error s (t) is respectively
E (t)=x-xr=[e1,e2,...,en]T∈Rn
S (t)=[KT 1]e
Wherein, xr=[x1r,x2r,...,xnr]TTo expect state vector, K=[k1,k2,...,kn-1]TIt is many for Hurwitz
Binomial coefficient, takes1≤j≤n-1, constant λ>0.
Step 3：Utilize OCBM biometrics design controller u：
For nonaffine controlled system as described above, design controller u is：
Controller u is by monitoring controller us, the asymptotic device u of biometricbio, and compensator uc(s, e) is constituted；Its specific table
Up to for：
uc(s, e)=- Kps-ηsat(s/β0)-Λ
Whereinc>0 is c (x) the known upper bound, control gain Kp>0, it is adaptive
Answer renewal rate r>0, filtering error s (t) decay with the time by exponential form, η>0 is Global Asymptotic precision, β1>0 is to allow row
For the cut off value with misdeed, β0>0 is the cut off value of acquired behavior and misdeed, is meeting β1>β0On the premise of, by setting
Meter person freely chooses；And using OCBM network structures as follows,
Wherein M is neural adaptive unit number, Gi(z) it is Gaussian function,For estimation
Weights, di(z)=[1, di,1,...,di,q]T=[1, z1-μi,1,...,zq-μi,q]T∈Rq+1Z is inputted to Gauss for current network
Function center μ Euclidean distance.And meet condition：The asymptotic error for making biometric model is
There is Global Asymptotic precision η>0 causes | ηi|≤η and | εbio|≤η；In addition, Weight number adaptively
Rule updates rule with parameter：
Wherein, ρ is weights learning speed, and I is (q+1) rank unit matrix, χ>0 is design constant.
Step 4：Controller u is applied to the Non-Affine Systems of step 2 foundation, makes output y (t) by given accuracy β0With
Track desired trajectory xd(t), while ensuring system tracking error e (t) in the bounded of t >=0.
Beneficial effects of the present invention：
The Neural Network Based Nonlinear control method of the imitative operant conditioned reflex of the present invention, its life from operant conditioned reflex
Thing principle is set out, and is constructed a kind of artificial neural network by bionical inspiration and is used it for handling complex uncertainty system
Unknown nonlinear, and design the control plan based on operant conditioned reflex model for the uncertain Non-Affine Systems of a class
Slightly so that such uncertain Non-Affine Systems control can reduce artificial tune ginseng work, independent of accurate drift nonlinear transformations,
Without off-line training with shutting down reprogram process, with more wide in range system operation conditions, and control essence can ensured
Less system operations resource is consumed while spending, validity, flexibility and the adaptivity of ANN Control is improved.
Brief description of the drawings
Fig. 1 is the basic theory of constitution schematic diagram of bionic neural network with neural adaptive unit；
Fig. 2 is the basic procedure schematic diagram for building i-th of neural adaptive unit；
Fig. 3 is the bionic neural network structure principle chart based on operant conditioned reflex bionic model (OCBM)；
Fig. 4 is using OCBM and filtering error evolution curve obtained by asymptotic control (SOAC) method of conventional ad-hoc
Figure；
Fig. 5 is using OCBM and control signal output evolution curve map obtained by SOAC methods；
Fig. 6 is using F0 system mode phase diagram curve maps obtained by OCBM methods
Fig. 7 is using F0 system mode phase diagram curve maps obtained by SOAC methods；
Fig. 8 is the number evolution curve map of neural adaptive unit.
Embodiment
The invention will be further described with reference to the accompanying drawings and examples.
The present embodiment imitates the Neural Network Based Nonlinear control method of operant conditioned reflex, including following not step：
Step 1: setting up operant conditioned reflex bionic model：
In the nervous system of higher organism, neuron inclusion is in nerve center region clustering formation nerve nucleus, function phase
As nerve nucleus set formed nerve nucleus.Inspired by this principle, the present invention is by the neuron in network by neururgic
Difference is classified, and makes to constitute a neural adaptive unit with identical neururgic neuron, universe network is by there is M god
Constituted through unit, then the nervous activity of i-th of neural unit：
Wherein, μi∈Rq, σi∈ R are the parameter of i-th of Neural cluster, z=[z1,z2,...,zq]TIt is defeated for neutral net
Enter.It can be seen that, neutral net cortex input z has to nervous activity to be directly affected.Work as μiDuring=z, Gi(z) maximum can be obtained.
Nervous activity principle, μ are maximized according to reward behavioriObtained by following formula：
Wherein, riAt the time of representing i-th of bonus event generation.In other words, if a certain behavior of individual makes its nerve
Activity is maximized, then dopamine neuron will record the behavior automatically and be stored in μiIn, and then pass through Gi(z) influence
Unit i output gi.By that analogy, after m-th bonus event occurs, Unit M will be activated, and form GM(z).Such as Fig. 1
Shown, with the generation of bonus event, Neural cluster is by sequential activation so that substantially oneself is presented in the overall structure of neutral net
Construction ability.
The detailed process of i-th new of neural adaptive unit of generation is：1) according to current Deviant behavior s (t), prize
Encourage marker position Reward_Flag and be assigned to true value；2) in (i-1) the individual unit generated, the nerve of all neurons
Activity meets 0<GSum<ω.Fig. 2 gives the automatic basic procedure for setting up Unit i.It can be seen that, by the operability for studying individual
Condition study mechanism is optimized with certain biorational to Traditional Man neuronal structure.
Fig. 3 gives the bionic neural network structure chart based on operant conditioned reflex model.The network is novel to be in
Biological nervous system can be simulated to a certain extent in learning period diencephalon inner structure automatically update process in it.Bonding behavior
The objective fact and discovery of psychology and neuro-physiology, by studying a series of changes that brain occurs during operant learning
Change, a kind of artificial neuron meta-model of bionical study is proposed exploratoryly.Specifically, the output g of (1) overall networkbioBy not
Output weighted sum with Unit is tried to achieve, rather than simple cumulative and form；(2) Unit create-rule is based on designed award
Strategy, Deviant behavior and the current nervous activity of system.A new Unit is often generated, (q+1) individual neuron just can be accordingly introduced
And (part as shown in phantom in Figure 3), therefore the introducing of useless/unrelated neuron is not resulted in, so as to save system operations money
Source and learning cost；(3) each Unit synaptic weight and basic function structural parameters is automatically updated in system operation,
And then avoid cumbersome artificial choosing ginseng and tune ginseng step.To sum up, compared to traditional artificial neural network, carrying model has relatively
Sound self study, adaptive and self-structuring ability.
Step 2: setting up the class Non-Affine Systems based on operant conditioned reflex bionic model：
Wherein, system mode vector x=[x1,...,xn]T∈Rn, system input/controller signals u ∈ R, system output
For y ∈ R；Fξ(t)() represents unknown nonlinear model of the structure with time drift, and concrete form is as follows：
Fξ(t)()={ Fi() | i=ξ (t) ∈ N+,0≤ti≤t<ti+1}
And Fξ(t)() meetsThere is bounded positive definite scalar function a (x), b (x), c (x) so that
|Fξ(t)(x, 0) |≤c (x),Set up.
Define system mode error vector e (t) and filtering error s (t) is respectively
E (t)=x-xr=[e1,e2,...,en]T∈Rn
S (t)=[KT 1]e
Wherein, xr=[x1r,x2r,...,xnr]TTo expect state vector, K=[k1,k2,...,kn-1]TIt is many for Hurwitz
Binomial coefficient, takes1≤j≤n-1, constant λ>0.
Step 3：Utilize OCBM biometrics design controller u：
For nonaffine controlled system as described above, design controller u is：
Controller u is by monitoring controller us, the asymptotic device u of biometricbio, and compensator uc(s, e) is constituted；Its specific table
Up to for：
uc(s, e)=α
Whereinc>0 is c (x) the known upper bound, control gain Kp>0, it is adaptive
Answer renewal rate r>0, filtering error s (t) decay with the time by exponential form, η>0 is Global Asymptotic precision, β1>0 is to allow row
For the cut off value with misdeed, β0>0 is the cut off value of acquired behavior and misdeed, is meeting β1>β0On the premise of, by setting
Meter person freely chooses；And using OCBM network structures as follows,
Wherein M is neural adaptive unit number, Gi(z) it is Gaussian function,For estimation
Weights, di(z)=[1, di,1,...,di,q]T=[1, z1-μi,1,...,zq-μi,q]T∈Rq+1Z is inputted to Gauss for current network
Function center μ Euclidean distance.And meet condition：The asymptotic error for making biometric model is
There is Global Asymptotic precision η>0 causes | ηi|≤η and | εbio|≤η；In addition, Weight number adaptively
Rule updates rule with parameter：
Wherein, ρ is weights learning speed, and I is (q+1) rank unit matrix, χ>0 is design constant, then has following closed loop control
System performance processed is set up：
(1) there is constant Tf>0 so that | s (t) |<β1,
(2) during system operation, | s (t) |>β0Total time TAIt is limited；
(3) as t → ∞, have | s (t) |≤β0And | ek(t)|≤2k-1λk-nβ0, k=1 ..., n；
Step 4：Controller u is applied to as above Non-Affine Systems, makes output y (t) by given accuracy β0Rail is expected in tracking
Mark xd(t), while ensuring system tracking error e (t) in the bounded of t >=0.
The validity for imitating the present embodiment the Neural Network Based Nonlinear control method of operant conditioned reflex below is imitated
True checking：
Consider following second order Non-Affine Systems：
Wherein, x=[x1,x2]T, Fξ(t)(x, u) is unknown drift nonlinear model.The present embodiment is used not to be floated with the time
The fixed structure system model of shifting, is contrasted with asymptotic control (SOAC) method of conventional ad-hoc, verifies having for OCBM methods
Effect property.
Given ideal trajectory xd(t)=3sin (0.1 π t), expectation state vectorInitially
State vector x (0)=[x1,x2]T=[2,3]T.Deviant behavior bound is set to β1=2, β0=0.03, control accuracy and β0
Value is identical.Control parameter is Kp=1, r=0.1.Intrinsic learning rate χ=0.5, weights learning speed ρ=8, excitement levels threshold
Value ω=0.1.Neural adaptive unit initial number M (0)=0, as M (t) >=1, newly-increased activation primitive width initial value is
σi=0.5, i=1 ..., M (t).In addition, reasonability and preciseness to ensure contrast, two methods control to increase using identical
Benefit and initiation parameter.The system emulation time is 80 seconds, sampling period Ts=10 milliseconds.
Make Fξ(t)(x, u)=F0(x, u) is set up to t >=0, and
F0(x, u)=3u+2sin (u)+0.5cos (x1+x2)
Formula usC=1 is taken in (s, Λ), it is known that met | F0(x,0)|≤c(x)<c。
Fig. 4 and Fig. 5 respectively depict the differentiation exported using filtering error obtained by OCBM and SOAC methods and control signal
Situation, two width subgraphs in it are respectively t ∈ [5,10] and t ∈ [58,60] amplification result.As seen from the figure, two kinds of controlling parties
Method can make filtering error with time Convergence.During whole system is run, conventional ad-hoc type controller can make filtering error
More shake is produced, and is carried in embodiment under OCBM controller actions, the change of filtering error and control action is then with respect to light
It is sliding.This also indicates that the biomimetic control based on OCBM has more preferable internal regulation ability, and it can avoid the high frequency for evoking system
Vibration, so that the service life of extension device.
Fig. 6, Fig. 7 successively give the system mode phase diagram based on two methods of OCBM and SOAC."×" represents ith
The corresponding system status information of reward behavior.What the solid line circle domain representation ANN training drawn by the center of circle of "×" was inputted compacts area
Domain, and each compact the unique neural adaptive unit (marked and distinguished by Unit in figure) of correspondence.In addition, biometric is asymptotic
Device ubioThe system mode justified to solid line in domain is effective, and justifying overseas state in solid line uses monitor usAnd compensator ucControlled
System.As can be seen that the running track of two methods, which can be tracked, gives ideal trajectory, and use OCBM methods can be with
Produce the neural adaptive unit that quantity is relatively fewer and area size can be automatically adjusted in real time.
Fig. 8 embodies neural adaptive unit number and changed with time situation.Notice after about 25 seconds, two kinds of controls
The Unit of method reaches stationary value and does not continue to increase.However, 23 Unit are ultimately generated using SOAC methods, and OCBM
Method only generates 9.It can be seen that, when performing same control task, system generation can be greatly decreased in the controller based on OCBM
Neuron population, so as to save system operations resource.
Finally illustrate, the above embodiments are merely illustrative of the technical solutions of the present invention and it is unrestricted, although with reference to compared with
The present invention is described in detail good embodiment, it will be understood by those within the art that, can be to skill of the invention
Art scheme is modified or equivalent substitution, and without departing from the objective and scope of technical solution of the present invention, it all should cover at this
Among the right of invention.
Claims (1)
1. a kind of Neural Network Based Nonlinear control method of imitative operant conditioned reflex, it is characterised in that：Comprise the following steps：
Step 1: setting up operant conditioned reflex bionic model：
Neuron in network is classified by neururgic difference, one is constituted with identical neururgic neuron
Neural adaptive unit, universe network is made up of M neural adaptive unit, then the nerve of i-th of neural adaptive unit is living
It is dynamic：
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Reward behavior maximizes nervous activity principle, μiObtained by following formula：
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Wherein, riAt the time of representing i-th of bonus event generation；
Step 2: setting up the class Non-Affine Systems based on operant conditioned reflex bionic model：
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Wherein, system mode vector x=[x1,...,xn]T∈Rn, system input/controller signals u ∈ R, system is output as y ∈
R；Fξ(t)() represents unknown nonlinear model of the structure with time drift, and concrete form is as follows：
Fξ(t)()={ Fi() | i=ξ (t) ∈ N+,0≤ti≤t<ti+1}
And Fξ(t)() meetsThere is bounded positive definite scalar function a (x), b (x), c (x) so that | Fξ(t)
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Define system mode error vector e (t) and filtering filtering error s (t) is respectively
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S (t)=[KT 1]e
Wherein, xr=[x1r,x2r,...,xnr]TTo expect state vector, K=[k1,k2,...,kn-1]TFor Hurwitz multinomials
Coefficient, takesConstant λ>0；
Step 3：Utilize OCBM biometrics design controller u：
For nonaffine controlled system as described above, design controller u is：
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For：
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1
Whereinc>0 is c (x) the known upper bound, control gain Kp>0, adaptively more
New speed r>0, filtering error s (t) decay with the time by exponential form, η>0 is Global Asymptotic precision, β1>0 be admissible action and
The cut off value of misdeed, β0>0 is the cut off value of acquired behavior and misdeed, is meeting β1>β0On the premise of, by designer
Freely choose；And use OCBM network structures as follows：
<mrow>
<msub>
<mover>
<mi>g</mi>
<mo>^</mo>
</mover>
<mrow>
<mi>b</mi>
<mi>i</mi>
<mi>o</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msup>
<mi>&Psi;</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>M</mi>
</munderover>
<msub>
<mover>
<mi>g</mi>
<mo>^</mo>
</mover>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>&Psi;</mi>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>G</mi>
<mrow>
<mi>S</mi>
<mi>u</mi>
<mi>m</mi>
</mrow>
</msub>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>M</mi>
</munderover>
<msub>
<mi>G</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mover>
<mi>g</mi>
<mo>^</mo>
</mover>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msubsup>
<mi>d</mi>
<mi>i</mi>
<mi>T</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<msub>
<mover>
<mi>W</mi>
<mo>^</mo>
</mover>
<mi>i</mi>
</msub>
<msub>
<mi>G</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
</mrow>
Wherein M is neural adaptive unit number, Gi(z) it is Gaussian function,To estimate weights,
di(z)=[1, di,1,...,di,q]T=[1, z1-μi,1,...,zq-μi,q]T∈Rq+1Z is inputted into Gaussian function for current network
Heart μ Euclidean distance；And meet condition：The asymptotic error for making biometric model is
<mrow>
<msub>
<mi>&epsiv;</mi>
<mrow>
<mi>b</mi>
<mi>i</mi>
<mi>o</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>g</mi>
<mrow>
<mi>b</mi>
<mi>i</mi>
<mi>o</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msup>
<mi>&Psi;</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>M</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</munderover>
<msubsup>
<mi>g</mi>
<mi>i</mi>
<mo>*</mo>
</msubsup>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
</mrow>
There is Global Asymptotic precision η>0 causes ηi|≤η and | εbio|≤η；In addition, Weight number adaptively rule with
Parameter updates rule：
<mrow>
<msub>
<mover>
<mover>
<mi>W</mi>
<mo>^</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>=</mo>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>&rho;Id</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<msub>
<mi>G</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<msup>
<mi>&Psi;</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>|</mo>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>|</mo>
<mo>></mo>
<msub>
<mi>&beta;</mi>
<mn>0</mn>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0</mn>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>o</mi>
<mi>t</mi>
<mi>h</mi>
<mi>e</mi>
<mi>r</mi>
<mi>w</mi>
<mi>i</mi>
<mi>s</mi>
<mi>e</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
<mrow>
<msub>
<mover>
<mi>&sigma;</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>=</mo>
<mo>-</mo>
<mi>&chi;</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>g</mi>
<mo>^</mo>
</mover>
<mrow>
<mi>b</mi>
<mi>i</mi>
<mi>o</mi>
</mrow>
</msub>
<mo>-</mo>
<msubsup>
<mi>d</mi>
<mi>i</mi>
<mi>T</mi>
</msubsup>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
<msub>
<mover>
<mi>W</mi>
<mo>^</mo>
</mover>
<mi>i</mi>
</msub>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mo>|</mo>
<mo>|</mo>
<mi>z</mi>
<mo>-</mo>
<msub>
<mi>&mu;</mi>
<mi>i</mi>
</msub>
<mo>|</mo>
<msup>
<mo>|</mo>
<mn>2</mn>
</msup>
</mrow>
<msubsup>
<mi>&sigma;</mi>
<mi>i</mi>
<mn>3</mn>
</msubsup>
</mfrac>
<mfrac>
<mrow>
<msub>
<mi>G</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>&Psi;</mi>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>;</mo>
</mrow>
Wherein, ρ is weights learning speed, and I is (q+1) rank unit matrix, χ>0 is design constant；
Step 4：Controller u is applied to step 2) set up Non-Affine Systems, make output y (t) by given accuracy β0The tracking phase
Hope track xd(t), while ensuring system tracking error e (t) in the bounded of t >=0.
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