CN109613826A - A kind of antihunt self-adaptation control method of fractional order arch MEMS resonator - Google Patents
A kind of antihunt self-adaptation control method of fractional order arch MEMS resonator Download PDFInfo
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Abstract
The present invention relates to a kind of antihunt self-adaptation control methods of fractional order arch MEMS resonator, belong to antihunt control field.In the design process, indeterminate is compensated using the Chebyshev neural network of single right value update, and solves the problems, such as that the control direction as caused by drive characteristic is unknown in Caputo fractional calculus using Nussbaum function.The Nonlinear Tracking Differentiator based on hyperbolic sine function is devised simultaneously, virtual controlling mid-score rank is solved and calculates complicated repetition differential problem.Then, under adaptive Reverse Step Control frame, using cline frequency distributed model, a kind of antihunt auto-adaptive control scheme merged with Nussbaum function, neural network and Nonlinear Tracking Differentiator has been invented.Based on fractional order Liapunov's stability criterion, it was demonstrated that the asymptotic stability of closed-loop system.Finally, the Simulation validity of proposed scheme.
Description
Technical field
The invention belongs to antihunt control field, the antihunt for being related to a kind of fractional order arch MEMS resonator is self-adaptive controlled
Method processed.
Background technique
In recent years, extension of the fractional calculus as integer rank calculus, in electronic engineering, robotics, biological work
The fields such as journey, signal processing cause the extensive concern of academia.It has robust ability, design freedom and mapping etc.
Potential superiority, can accurate description Practical Project object and technical process, be widely used in control system.Micro-electro-mechanical systems
System (MEMS) resonator is received significant attention due to being widely applied with sensor, micro-valve, switch and filter etc..MEMS
The nonlinear characteristic that there is resonator parallel-plate to force, middle plane and squeeze-film damping are contour.These characteristics may cause chaos
Oscillation, this is unwelcome, and may cause adverse reaction.Simultaneously as the variation of external environment and depositing for manufacturing defect
Needing to face the new challenge such as fluctuation, mechanical couplings and hazardous noise of characteristic parameter.As fractional calculus exists
Universal and system performance high quality requirement in engineering, the antihunt self adaptive control etc. of fractional order arch form MEMS resonator are opened
Putting property problems demand solves.
In order to stablize the unstable periodic orbits of chaos system, Ott, Grebogi and Yorke etc. first proposed the side OGY
Method.Later, for integer level system synchronization and chaos controlling problem, researcher propose self adaptive control, Reverse Step Control,
The stable effective ways of many realization systems such as sliding formwork control, H ∞ control and contraction theory.Unfortunately, these methods only limit
It unites in the integer level of not drive characteristic, arch MEMS resonator can be directly applied to and need further to study.It utilizes
Fractional calculus theory has carried out more accurate modeling to chaos system.For a long time, chaotic systems with fractional order such as Cai Shi electricity
Road, Rssler system, L ü system and Van Der Pol Duffing system all have been reported that in fractional calculus field, but without reference to
Incentives plus restraints.In practical projects, the drive characteristic including input dead zone and saturation is inevitable.Ignore this characteristic
It will lead to that system is unstable or penalty.
Some researchers discuss the torsion MEMS micromirror Second Order Sliding Mode Control scheme of the 2D with side-wall electrode, the control
Device is made of equivalent control and switch control, to solve the uncertainty and external disturbance of model.Sa Lafu etc. is by band logical SMC skill
Art is applied to the drive mode of MEMS resonator and resonant frequency sensor.But intrinsic shake relevant to SMC cannot quilt completely
Inhibit.Meanwhile these researchers are not dedicated to fractional calculus and drive characteristic problem in MEMS system.Contragradience control
System is one of the effective tool of fractional order nonlinear system controller design.Fourth etc. solves parameter not under sublevel inverting framework
Know, the pseudo- state Stabilization of the fractional order nonlinear system that matches with additional disturbance.Das and Yadav are ground by Backstepping
The chaos controlling and Function Projective Synchronization problem of fractional order T system and Lorenz chaos system are studied carefully.Weis etc. are directed to non-linear point
Number level system, proposes a kind of output feedback ontrol method based on adaptive contragradience.These methods depend critically upon accurately
System modelling cannot handle nonlinear function unknown in dynamic model.It is relevant to contragradience with the increase of system divisor
" system explosion " phenomenon will occur.
Summary of the invention
In view of this, the purpose of the present invention is to provide a kind of antihunt of fractional order arch MEMS resonator is self-adaptive controlled
Method processed.
In order to achieve the above objectives, the invention provides the following technical scheme:
A kind of antihunt self-adaptation control method of fractional order arch MEMS resonator, method includes the following steps:
S1: utilizing Galerkin decomposition method, establishes the fractional order arch form MEMS resonator with unknown drive characteristic
Kinetic model；
S2: design adaptive controller.
Further, the step S1 specifically:
Using Galerkin decomposition method, by the dynamics of the fractional order arch form MEMS resonator with unknown drive characteristic
Model is write as
Wherein variable-definition are as follows:
Indicate ratio,It is nondimensional time variable,Indicate damped coefficient,Indicate extensograph parameter,It is voltage parameter,Indicate frequency, h=h0/g0It indicates just
Beginning rising value,It is constant, x1=q (t) indicates displacement,Indicate speed, α indicates fractional order, C
Indicate that the symbol that Caputo is defined in Fractional Calculus field, M (u) indicate unknown drive characteristic,Indicate length coordinate,
Indicate first normalization mode,Indicate amount of deflection,Indicate characteristic,Indicate the time,Indicate that constant, u indicate practical control
System input,kc1> 0；L is length, and A is cross-sectional area；B is width, CvFor viscous damping system
Number, d is thickness,For Young's modulus, IyFor rotary inertia, ρ is mass density, Ω0For harmonic load frequency, εa0For vacuum Jie
Electric constant, VDCFor DC voltage, VACFor alternating voltage, w0For arch displacement, ω0For driving frequency；
There are asymmetric Non-smooth surface saturation nonlinearity drive characteristic M (u) in input, it is expressed as
WhereinWithηIndicate boundary, a1(t) and a2(t) time-varying function, l are indicated1And l2Indicate Dead Zone function,WithIndicate positive unknown breakpoint；
Due to a1(t) and a2(t) it is time-varying, and asymmetric being saturated Non-smooth surface, it is asymmetric to approach introduces smooth function
Non-smooth surface saturation characteristics
M (u)=S (u)+D (u) (3)
With
Wherein w indicates that design parameter, D (u) indicate approximate error and have | D (u) |=M (u)-S (u)≤Γ, Γ are indicated just
Determine unknown constant；
Had according to mean value theorem for smooth function S (u)
DefinitionWith obtain S (0)=0, then (3) are rewritten as M (u)=pu+D (u) (6)
a1(t) and a2(t) it indicates time-varying function, practical feelings of the nonlinear system when being interfered by inside and outside can be reacted
Condition；This sliding function only needs the bound of drive characteristic；The value different as Coefficients of Approximation w causes to approach the difference of M (u)
As a result；
System parameter selection is γ=7.993, h=0.3, μ=0.1, α=0.98, β=119.9883 and ω0=
0.4706；With the help of variable step START/TR BDF2 solver, disclosed by different fractional order and driving amplitude
The chaotic oscillation of fractional order arch MEMS resonator；Transient Chaos are as α=1.0 and 0.95 present in fractional order value；
Then arch MEMS resonator is suddenly switched to non-chaotic state at α=0.9 and 0.75；
Define 1: the Caputo definition of f (t) is expressed as in fractional derivative
WhereinIndicate gamma function, n and f(n) (t) indicate integer and f (t) n order derivative；
Lemma 1: for continuous functionWithFollowing equation is set up
Wherein 0 < α < 1；
Utilize lemma 1 and relational expressionIt obtains:
Wherein
Lemma 2: new fractional-order systemAnd 0 < α < 1, it willIt is transformed to fractional order product
The LINEAR CONTINUOUS frequency distribution model of point device is
WhereinIndicate weighting function,The time of day of expression system；
Define 2: if function N (η) meets with properties
It is referred to as Nussbaum function；
Nussbaum function is considered as the effective tool for handling drive characteristic unknown symbols problem；Introduce it is following with
The lemma of Nussbaum functional dependence, in order to controller design and stability analysis；
Lemma 3: assuming that V () and η () be [0 ∞) and have the smooth function of V (τ) >=0, N () is Nussbaum
Function, inequality below are set up
Wherein C0> 0, g (t) are non-zero constants,Indicate suitable constant, then V (t), η (t) and
It is bounded；
Assuming that 1: reference locus xdAnd its n order derivative is known and bounded；Meanwhile state variable x1(t) and x2(t) may be used
Measurement；
For the fractional order arch form MEMS resonator with uncertain and time-varying drive characteristic, propose a kind of self-adaptive controlled
Scheme processed, so that output y=x1(t) reference locus x is followed to slight errord, while it is related to chaotic behavior and asymmetric dead zone
Oscillation be totally constrained.
Further, the step S2 specifically:
Chebyshev multinomial is with the formal character of two recurrence formula
Ti+1(X)=2XTi(X)-Ti-1(X),T0(X)=1 (14)
Wherein X ∈ R and T1(X) it is defined as X, 2X, 2X-1 or 2X+1；
For [x1,…,xm]T∈Rm, a kind of polynomial strengthening form of Chebyshev is built as ξ (X)=[1, T1
(x1),…,Tn(x1),…,T1)xm),…,Tn)xm)] (15)
Wherein Ti)xj), i=1 ..., n, j=1 ..., m indicate that Chebyshev multinomial, ξ (X) indicate that Chebyshev is more
The basis function vector of item formula, n indicate order；
For compacting upper any given unknown continuous function f (X),Based on the general of Chebyshev neural network
Approximation theory adequately accurately approaches it, has
Wherein φ (t) is smooth weight vector；
There are Chebyshev neural networks
Wherein ε (X) > 0 is approximate error, ΩφAnd DXRespectively indicate the appropriate sets with compact border of φ (t) He X；If optimal ginseng
Number φ*It is equal toWherein φ*Referred to as artificial amount；WhenShi You
To promote to take such as down conversion quickly in line computation and reduce Chebyshev neural network weight vector number
WhereinRelational expressionDeposit it is vertical,It is λi(t) estimated value, biIt is one
Small normal number；
By Young inequality, a mathematic(al) manipulation related with weight vector number of Chebyshev neural network is exported；
Step 1: defining first intermediate variable
Wherein tracking error e1(t) it is defined as e1(t)=x1(t)-xd(t), σ1Indicate positive design parameter；If Z1(t)→
0,So e1(t) → 0 He
Selection second has tracking errorIntermediate variable e2(t)=x2(t)-α2(t),
Wherein σ2> 0 is a design parameter, α2(t) virtual controlling is indicated；Z is derived in the definition of Caputo fractional calculus1
(t) derivative
Virtual controlling is selected as
Wherein k1> 0 represents control gain；
Based on lemma 1, following cline frequency distributed model is obtained:
Consider Liapunov stability criterion
Wherein
To V1(t) seeking time derivative
Step 2: selection fractional order Liapunov stability criterion
Wherein h2> 0,
To Z2(t) differential obtains
Wherein
f2() is a high-order nonlinear function, wherein such as h, β, γ and b11Etc. system parameters cannot accurately measure,
And it is influenced to establish accurate system model by internal and external factors extremely difficult；Different external drives is humorous to arch MEMS
Vibration device can generate harmful oscillation, this to vibrate the performance that reduce system to a certain extent；To solve these problems, it uses
Chebyshev neural network
In fact, cannot directly be found out due to calculating complexityTo solve this problem, design is being based on hyperbolic just
The fractional order Nonlinear Tracking Differentiator of string function estimates virtual controlling α2(t) Fractional Derivative
The wherein Nonlinear Tracking Differentiator state z based on hyperbolic sine function2,2WithIt is equal, r2> 0, ci> 0, i=
1,2 and di> 0, i=1,2 is design constant, exist be with T positive number relational expression
(27) and (28) are substituted into (26), are obtained
It is easy exportUsing lemma 1, cline frequency distributed model is further derived
Take the time-derivative of (25)
Wherein
It is inputted using being controlled below Nussbaum construction of function
Wherein k21> 0 and k22> 0 is control gain, has more new law
Wherein g2It is positive number；
More new law and control law substitute into (31), acquire
Theorem 1: assuming that 1 deposit vertical under conditions of, consider the sublevel arch MEMS resonator with unknown drive characteristic, such as
Fruit proposes antihunt self-adaptation control method (32) intervention being made of adaptive rate (33) and (34), then all internal signals
Bounded is kept, while being completely eliminated comprising the oscillation including chaotic behavior and asymmetric dead zone；
It proves: defining entire Liapunov candidate functions
By?
Wherein
DefinitionAbove formula is reduced to
Above formula both sides are multiplied simultaneouslyIt obtains
DefinitionAbove formula is integrated
Z1(t), Z2(t) andBelong to and compacts
Therefore, all signals in closed-loop system are all bounded；Further prove
So far, the proof of theorem 1 is completed.
The beneficial effects of the present invention are:
1) for the uncertain dynamics and perturbed problem in Caputo fractional calculus field, a kind of update is proposed
Single weight Chebyshev neural network.It can promote quickly to reduce the requirement to dynamic control equation in line computation.
It solves to control direction uncertain problem as caused by drive characteristic using Nussbaum function, preferably avoids control input
Vibration in asymmetric dead zone threshold value.
2) devise a kind of Nonlinear Tracking Differentiator based on hyperbolic sine function, overcome in traditional contragradience technology " item is quick-fried
It is fried " problem.Compared with firstorder filter and general Nonlinear Tracking Differentiator, the estimated accuracy of virtual controlling differential term is higher.
3) it using the principle of the adaptive Backstepping based on cline frequency distributed model, proposes to combine Nonlinear Tracking Differentiator, mind
Antihunt auto-adaptive control scheme through network and Nussbaum function forces system mode to be approached with minimum error with reference to letter
Number.Meanwhile for fractional order arch form MEMS resonator, resonance frequency is realized in the practical control within asymmetric dead zone threshold value
Vibrationproof purpose near rate reduces controller vibration.
Detailed description of the invention
In order to keep the purpose of the present invention, technical scheme and beneficial effects clearer, the present invention provides following attached drawing and carries out
Illustrate:
Fig. 1 is the schematic diagram of arch MEMS resonator；
Fig. 2 is the drive characteristic influenced by asymmetric Non-smooth surface saturation nonlinearity；
Fig. 3 is the phasor under different fractional orders；It (a) is 1.0；It (b) is 0.95；It (c) is 0.9；It (d) is 0.75；
Fig. 4 is the phasor under different excitation amplitudes；It (a) is 0.01；It (b) is 0.02；It (c) is 0.1；It (d) is 0.21；
Fig. 5 is position tracking；
Fig. 6 is speed tracing；
Fig. 7 is the control input before and after drive characteristic；
Fig. 8 is the Nussbaum function under different excitation amplitudes；
Fig. 9 is the intermediate variable under different excitation amplitudes；It (a) is the intermediate variable Z under different excitation amplitudesi(t), i=
1；It (b) is the intermediate variable Z under different excitation amplitudesi(t), i=2；
Figure 10 is the control input under different excitation amplitudes and fractional order；It (a) is the control under different excitation amplitudes
Input；It (b) is the control input under different fractional orders；
Figure 11 is that single weight is updated under different excitation amplitudes and fractional order；(a) under different excitation amplitudes more
New list weight；It (b) is the single weight of update under different fractional orders；
Figure 12 is the phasor under different excitation amplitudes and fractional order；It (a) is the phasor under different excitation amplitudes；(b)
For the phasor under different fractional orders；
Figure 13 is the intermediate variable under different fractional order values；
Figure 14 is the tracking mode z under different excitation amplitudes and fractional order2,2；(a) for the anti-of different excitation amplitudes
Vibrate performance；It (b) is the antihunt performance to different fractional orders；
Figure 15 is the control input under different schemes；
Figure 16 is state z of the Nonlinear Tracking Differentiator under different schemes2,2；
Figure 17 is the second intermediate variable under different schemes.
Specific embodiment
Below in conjunction with attached drawing, a preferred embodiment of the present invention will be described in detail.
1. problem proposes and Fundamentals of Mathematics
Arch MEMS resonator holds arch by direct current (DC), alternating current (AC), hearth electrode, double fastener and sharpens Wei Liang, two anchors
It is formed with integrated operational amplifier.Obviously, the fundamental resonance frequency of this arch MEMS resonator is higher than fixed beam or cantilever beam.
Fig. 1 illustrates the schematic diagram of the arch MEMS resonator of electrostatic drive.When clearance distance is g0And it is parallel to the bottom electricity of X-axis placement
When pole works, electrostatic drive is executed.Amount with physical size is expressed asUsing Galerkin decomposition method, will have unknown
The kinetic model of the fractional order arch form MEMS resonator of drive characteristic is write as
Wherein variable-definition is
Indicate ratio,It is nondimensional time variable,Indicate damped coefficient,Indicate extensograph parameter,It is voltage parameter,Indicate frequency, h=h0/g0It indicates just
Beginning rising value,It is constant, x1=q (t) indicates displacement,Indicate speed, α indicates fractional order, C
Indicate that the symbol that Caputo is defined in Fractional Calculus field, M (u) indicate unknown drive characteristic,Indicate length coordinate,
Indicate first normalization mode,Indicate amount of deflection,Indicate characteristic,Indicate the time,Indicate that constant, u indicate practical control
System input,kc1> 0.
Remaining pa-rameter symbols of arch MEMS resonator are given in Table 1.
The expression of 1 system parameter of table
In engineering practice, drive characteristic is inevitable.Its appearance can cause inaccurate or systematic jitters.
There are asymmetric Non-smooth surface saturation nonlinearity drive characteristic M (u) in input, it is expressed as
WhereinWithηIndicate boundary, a1(t) and a2(t) time-varying function, l are indicated1And l2Indicate Dead Zone function,WithIndicate positive unknown breakpoint.
The structure of Fig. 2 expression drive characteristic.Due to a1(t) and a2(t) it is time-varying, and asymmetric being saturated Non-smooth surface, makes
The design for obtaining controller is extremely difficult.For this problem, smooth function is introduced to approach asymmetric Non-smooth surface saturation characteristics
M (u)=S (u)+D (u) (3)
With
Wherein w indicates that design parameter, D (u) indicate approximate error and have | D (u) |=M (u)-S (u)≤Γ, Γ are indicated just
Determine unknown constant.
Had according to mean value theorem for smooth function S (u)
DefinitionWith obtain S (0)=0, then (3) are rewritten as M (u)=pu+D (u) (6)
Remarks 1:a1(t) and a2(t) time-varying function is indicated, they can effectively react nonlinear system by inside and outside
Actual conditions when portion interferes.This sliding function only needs the bound of drive characteristic.The value different as Coefficients of Approximation w can be with
Lead to the different Approaching Results to M (u), in this way more suitable for real system.
Fig. 3 is the phasor under different fractional orders；It (a) is 1.0；It (b) is 0.95；It (c) is 0.9；It (d) is 0.75；Fig. 4 is not
With the phasor under excitation amplitude；It (a) is 0.01；It (b) is 0.02；It (c) is 0.1；It (d) is 0.21；System parameter selection be γ=
7.993, h=0.3, μ=0.1, α=0.98, β=119.9883 and ω0=0.4706.It is asked in variable step START/TRBDF2
With the help of solving device, fractional order arch MEMS resonator is disclosed by physical conditions such as different fractional orders and driving amplitude
Chaotic oscillation.In all cases, fractional order arch MEMS resonator all shows the unstable attraction with chaotic transience
Son.In Fig. 3, Transient Chaos are as α=1.0 and 0.95 present in fractional order value.Then arch MEMS resonator exists
Non-chaotic state is suddenly switched at α=0.9 and 0.75.Fig. 4 shows that excitation amplitude variation causes different chaotic motions.By
Has the characteristics that randomness and unpredictability in chaotic oscillation, if taking no action to overcome the property that will necessarily reduce system
Energy.
Define 1: the Caputo definition of f (t) can be expressed as in fractional derivative
WhereinIndicate gamma function, n and f(n)(t) the n order derivative of integer and f (t) is indicated.
Lemma 1: for continuous functionWithFollowing equation is set up
Wherein 0 < α < 1.
Utilize lemma 1 and relational expressionIt is available:
Wherein
Lemma 2: new fractional-order systemAnd 0 < α < 1, it willIt is transformed to fractional order product
The LINEAR CONTINUOUS frequency distribution model of point device is
WhereinIndicate weighting function,The time of day of expression system.
Define 2: if function N (η) meets with properties
It is referred to as Nussbaum function.
Nussbaum function is considered as the effective tool for handling drive characteristic unknown symbols problem.Introduce it is following with
The lemma of Nussbaum functional dependence, in order to controller design and stability analysis.
Lemma 3: assuming that V () and η () be [0 ∞) and have the smooth function of V (τ) >=0, N () is Nussbaum
Function, inequality below are set up
Wherein C0> 0, g (t) are non-zero constants,Indicate suitable constant, then V (t), η (t) and
It is bounded.
Assuming that 1: reference locus xdAnd its n order derivative is known and bounded.Meanwhile state variable x1(t) and x2(t) may be used
It can measure.
Control target of the invention is for the fractional order arch form MEMS resonant with uncertain and time-varying drive characteristic
Device proposes a kind of auto-adaptive control scheme, so that output y=x1(t) reference locus x can with slight error be followedd, while with
The relevant oscillation of chaotic behavior and asymmetric dead zone is totally constrained.
2. adaptive controller designs
Chebyshev neural network has powerful function learning and approximation capability, is widely used in nonlinear system
Control and modeling.It is used to processing unknown function.Chebyshev multinomial is with the formal character of two recurrence formula
Ti+1(X)=2XTi(X)-Ti-1(X),T0(X)=1 (14)
Wherein X ∈ R and T1(X) X, 2X, 2X-1 or 2X+1 are generally defined as.
For [x1,…,xm]T∈Rm, a kind of polynomial strengthening form of Chebyshev is built as
ξ (X)=[1, T1(x1),…,Tn(x1),…,T1(xm),…,Tn(xm)] (15)
Wherein Ti(xj), i=1 ..., n, j=1 ..., m indicate that Chebyshev multinomial, ξ (X) indicate that Chebyshev is more
The basis function vector of item formula, n indicate order.
For compacting upper any given unknown continuous function f (X),It can be based on Chebyshev neural network
General approximation theory adequately accurately approaches it, has
Wherein φ (t) is smooth weight vector.
There are Chebyshev neural networks
Wherein ε (X) > 0 is approximate error, ΩφAnd DXRespectively indicate the appropriate sets with compact border of φ (t) He X.If optimal ginseng
Number φ*It is equal toWherein φ*Referred to as artificial amount.WhenShi You
Chebyshev neural network weight vector number is reduced in order to promote quickly to take in line computation such as down conversion
WhereinRelational expressionDeposit it is vertical,It is λi(t) estimated value, biIt is one
A small normal number.
Remarks 2: by means of Young inequality, one for being derived Chebyshev neural network is related with weight vector number
Mathematic(al) manipulation.This conversion can accelerate the speed of line solver, reduce difficulty, because it only needs a weight.
Step 1: defining first intermediate variable
Wherein tracking error e1(t) it is defined as e1(t)=x1(t)-xd(t), σ1Indicate positive design parameter.If Z1(t)→
0,So e1(t) → 0 He
Selection second has tracking errorIntermediate variable e2(t)=x2(t)-α2(t),
Wherein σ2> 0 is a design parameter, α2(t) virtual controlling is indicated.Z is derived in the definition of Caputo fractional calculus1
(t) derivative
Virtual controlling is selected as
Wherein k1> 0 represents control gain.
Based on lemma 1, following cline frequency distributed model is obtained:
Consider Liapunov stability criterion
Wherein
To V1(t) seeking time derivative
Step 2: selection fractional order Liapunov stability criterion
Wherein h2> 0,
To Z2(t) differential obtains
Wherein
Obviously, f2() is a high-order nonlinear function, wherein such as h, β, γ and b11Etc. system parameters cannot be accurate
Measurement, and by internal and external factors influenced to establish accurate system model extremely difficult.In addition, different external drives pair
Arch MEMS resonator can generate harmful oscillation, this to vibrate the performance that reduce system to a certain extent.In order to solve
These problems use Chebyshev neural network
In fact, cannot directly be found out due to calculating complexityIn order to solve this problem, it devises based on double
The fractional order Nonlinear Tracking Differentiator of bent SIN function estimates virtual controlling α2(t) Fractional Derivative
The wherein Nonlinear Tracking Differentiator state z based on hyperbolic sine function2,2WithIt is equal, r2> 0, ci> 0, i=
1,2 and di> 0, i=1,2 is design constant, exist be with T positive number relational expression
Remarks 3:
1) compared with traditional contragradience algorithm, the Nonlinear Tracking Differentiator proposed by the present invention based on hyperbolic sine function can be solved
Certainly complexity growing concern.Meanwhile it is able to solve the problem of firstorder filter low precision relevant to dynamic surface control, and
And it is suitable for that there is arbitrary input αi(t) system.
2) summarize the adjusting rule of fractional order Nonlinear Tracking Differentiator parameter in practical applications.(a)r2、c1And d1Directly determine
Convergence rate and precision, but excessive c1And d1It can cause overshoot.(b)c2And d2Also convergence rate and precision are influenced, so
And too small c2And d2It will lead to undesirable overshoot.
(27) and (28) are substituted into (26), are obtained
It is easy exportUsing lemma 1, cline frequency distributed model can be further derived
Take the time-derivative of (25)
Wherein
It is inputted using the following control of Nussbaum construction of function
Wherein k21> 0 and k22> 0 is control gain, has more new law
Wherein g2It is positive number.
More new law and control law substitute into (31), acquire
Theorem 1: assuming that 1 deposit vertical under conditions of, consider the sublevel arch MEMS resonator with unknown drive characteristic, such as
Fruit proposes antihunt self-adaptation control method (32) intervention being made of adaptive rate (33) and (34), then all internal signals
Bounded is kept, while being completely eliminated comprising the oscillation including chaotic behavior and asymmetric dead zone.
It proves: defining entire Liapunov candidate functions
ByIt can obtain
Wherein
DefinitionAbove formula can be reduced to
Above formula both sides are multiplied simultaneouslyIt obtains
DefinitionAbove formula is integrated
Z1(t), Z2(t) andBelong to and compacts
Therefore, all signals in closed-loop system are all bounded.Particularly, it can further be proved
Up to the present, the proof of theorem 1 has been completed.
3. interpretation of result
The validity just suggested plans is tested by analysis of simulation experiment.Select time-varying reference locus for xd=0.16sin
(2.5t).The problem of in order to handle drive characteristic unknown symbols, selects Nussbaum functionTo meet phase
The property answered.In order to overcome jitter phenomenon, sign () is replaced with arctan (10) here.Score based on hyperbolic sine function
The parameter selection of rank Nonlinear Tracking Differentiator is r2=9, c1=12, c2=0.2, d1=2 and d2=6.
For the ease of controller design, introduces smooth function S (u) and come the unknown drive characteristic of close approximation, function ginseng
Number is η=-0.1 and w=0.2.Selection control parameter is k1=12, k21=20, k22=20, h2=8, b2=0.2, g2
=2, σ1=0.3 and σ2=0.3.The initial value of all variables is set as zero.In addition, using single layer Chebyshev neural network,
And it devises Chebyshev polynomial basis function and is
Fig. 5 is position tracking；Fig. 6 is speed tracing；Fig. 5-6 discloses blue solid lines and red dotted line in a short period of time
It is completely overlapped.Simulation result shows program tracking accuracy with higher and faster convergence rate.
Fig. 7 describes fractional order arch MEMS resonator by being saturated the control constituted under unknown drive characteristic with Dead Zone
System input.It is worth noting that, the controller can be effectively prevented from the oscillation of control input in asymmetric dead zone threshold value.
In the case where taking no action to, the indefinite deterioration that can cause controller performance in the direction of drive characteristic.Fig. 8 is disclosed
Three curves of Nussbaum function are consistent under different excitation amplitudes.It can be concluded that drive characteristic direction is indefinite
Problem has obtained very good solution herein.Fig. 9 illustrates the intermediate variable Z under different excitation amplitudes with integral termi
(t), i=1,2.All curves are overlapped and converge to zero neighborhood, while demonstrating proposed scheme and taking the photograph with good parameter
Kinetic force.Fig. 9 (a) is the intermediate variable Z under different excitation amplitudesi(t), i=1；Fig. 9 (b) be different excitation amplitudes under in
Between variable Zi(t), i=2；
The more stringent condition such as drive characteristic, chaotic oscillation, model uncertainty is considered, although not due to model
Certainty, excitation amplitude and the fractional order difference of system, but Figure 10 illustrates in practical control without there is asymmetric dead zone threshold value
Interior oscillation phenomenon.In addition, control input also achieves no flutter even if whole system faces exacting terms.Figure 10 be
Control input under difference excitation amplitude and fractional order；It (a) is the control input under different excitation amplitudes；(b) in difference
Control input under fractional order；
Figure 11 is that single weight is updated under different excitation amplitudes and fractional order；(a) under different excitation amplitudes more
New list weight；It (b) is the single weight of update under different fractional orders；The update list weight of Chebyshev neural networkDirectly
Connect the approximation capability for influencing high-order nonlinear function.Fractional calculus has dynamic modeling and nonlinear Control ability, can
The dynamic characteristic of system is more accurately described.However fractional order value can cause chaotic oscillation and controller flutter.Figure 11 is illustrated
Chaotic oscillation and controller flutter have obtained very good solution herein.
Figure 12 is the phasor under different excitation amplitudes and fractional order；It (a) is the phasor under different excitation amplitudes；(b)
For the phasor under different fractional orders；Compared with Fig. 3-4, Figure 12 discloses fractional order arch MEMS resonator and is switched to a rule
Motion state, and intrinsic chaotic oscillation is totally constrained.The result of Figure 13 illustrates intermediate variable curve to fractional order
Change insensitive.
Figure 14 is the tracking mode z under different excitation amplitudes and fractional order2,2；(a) for the anti-of different excitation amplitudes
Vibrate performance；It (b) is the antihunt performance to different fractional orders；Hyperbolic sine function is considered as the ideal choosing of Nonlinear Tracking Differentiator
It selects.Because sinh () becomes linearly when close to 0, and when far from 0, sinh () becomes non-linear.On the one hand,
The nonlinear characteristic can eliminate buffeting with fast convergence, linear characteristic.On the other hand, it directly seeks by complicated calculationsIt is very difficult.Figure 14 show the Nonlinear Tracking Differentiator have excellent tracing function approximation capability with to Bu Tong sharp
Encourage the antihunt performance of amplitude and fractional order.
In order to further illustrate the superiority of Nonlinear Tracking Differentiator (HSF) scheme used based on hyperbolic sine function, here
It introduces Nonlinear Tracking Differentiator (NTD) and Nonlinear Tracking Differentiator (SMTD) two schemes based on sliding formwork compares.
Figure 15 illustrates that SMTD has the smallest flutter in 0.5 second, and the control of three schemes inputs the base after 0.5 second
This is equal.However, to show that the scheme (blue line) invented due to the smallest shake and amplitude is substantially better than NTD (red by Figure 16-17
Color dotted line) and SMTD (green dotted line).
Finally, it is stated that preferred embodiment above is only used to illustrate the technical scheme of the present invention and not to limit it, although logical
It crosses above preferred embodiment the present invention is described in detail, however, those skilled in the art should understand that, can be
Various changes are made to it in form and in details, without departing from claims of the present invention limited range.
Claims (3)
1. a kind of antihunt self-adaptation control method of fractional order arch MEMS resonator, it is characterised in that: this method include with
Lower step:
S1: utilizing Galerkin decomposition method, establishes the power with the fractional order arch form MEMS resonator of unknown drive characteristic
Learn model；
S2: design adaptive controller.
2. a kind of antihunt self-adaptation control method of fractional order arch MEMS resonator according to claim 1, special
Sign is: the step S1 specifically:
Using Galerkin decomposition method, by the kinetic model of the fractional order arch form MEMS resonator with unknown drive characteristic
It is write as
Wherein variable-definition are as follows:
Indicate ratio,It is nondimensional time variable,Indicate damped coefficient,Indicate extensograph parameter,It is voltage parameter,Indicate frequency, h=h0/g0It indicates just
Beginning rising value,It is constant, x1=q (t) indicates displacement,Indicate speed, α indicates fractional order, C
Indicate that the symbol that Caputo is defined in Fractional Calculus field, M (u) indicate unknown drive characteristic,Indicate length coordinate,
Indicate first normalization mode,Indicate amount of deflection,Indicate characteristic,Indicate the time,Indicate that constant, u indicate practical control
System input,kc1> 0；L is length, and A is cross-sectional area；B is width, CvFor viscous damping system
Number, d is thickness,For Young's modulus, IyFor rotary inertia, ρ is mass density, Ω0For harmonic load frequency, εa0For vacuum Jie
Electric constant, VDCFor DC voltage, VACFor alternating voltage, w0For arch displacement, ω0For driving frequency；
There are asymmetric Non-smooth surface saturation nonlinearity drive characteristic M (u) in input, it is expressed as
WhereinWithηIndicate boundary, a1(t) and a2(t) time-varying function, l are indicated1And l2Indicate Dead Zone function,WithTable
Show positive unknown breakpoint；
Due to a1(t) and a2(t) it is time-varying, and asymmetric being saturated Non-smooth surface, introduces smooth function to approach asymmetric non-light
Sliding saturation characteristics
M (u)=S (u)+D (u) (3)
With
Wherein w indicates that design parameter, D (u) indicate approximate error and have | D (u) |=M (u)-S (u)≤Γ, Γ indicate positive definite not
Know constant；
Had according to mean value theorem for smooth function S (u)
DefinitionWith obtain S (0)=0, then (3) are rewritten as
M (u)=pu+D (u) (6)
a1(t) and a2(t) it indicates time-varying function, actual conditions of the nonlinear system when being interfered by inside and outside can be reacted；This
The sliding function of kind only needs the bound of drive characteristic；The value different as Coefficients of Approximation w leads to the different Approaching Results to M (u)；
System parameter selection is γ=7.993, h=0.3, μ=0.1, α=0.98, β=119.9883 and ω0=0.4706；?
With the help of variable step START/TRBDF2 solver, fractional order arch is disclosed by different fractional order and driving amplitude
The chaotic oscillation of MEMS resonator；Transient Chaos are as α=1.0 and 0.95 present in fractional order value；Then arch
MEMS resonator is suddenly switched to non-chaotic state at α=0.9 and 0.75；
Define 1: the Caputo definition of f (t) is expressed as in fractional derivative
WhereinIndicate gamma function, n and f(n)(t) the n order derivative of integer and f (t) is indicated；
Lemma 1: for continuous function f1 *(t) andFollowing equation is set up
Wherein 0 < α < 1；
Utilize lemma 1 and relational expressionIt obtains:
Wherein
Lemma 2: new fractional-order systemAnd 0 < α < 1, it willIt is transformed to fractional order integrator
LINEAR CONTINUOUS frequency distribution model be
WhereinIndicate weighting function,The time of day of expression system；
Define 2: if function N (η) meets with properties
It is referred to as Nussbaum function；
Nussbaum function is considered as the effective tool for handling drive characteristic unknown symbols problem；Introduce following and Nussbaum
The lemma of functional dependence, in order to controller design and stability analysis；
Lemma 3: assuming that V () and η () be [0 ∞) and have the smooth function of V (τ) >=0, N () is Nussbaum letter
Number, inequality below are set up
Wherein C0> 0, g (t) are non-zero constants,Indicate suitable constant, then V (t), η (t) andIt is to have
Boundary；
Assuming that 1: reference locus xdAnd its n order derivative is known and bounded；Meanwhile state variable x1(t) and x2(t) it can survey
Amount；
For the fractional order arch form MEMS resonator with uncertain and time-varying drive characteristic, a kind of self adaptive control side is proposed
Case, so that output y=x1(t) reference locus x is followed to slight errord, while vibration relevant to chaotic behavior and asymmetric dead zone
It swings and is totally constrained.
3. a kind of antihunt self-adaptation control method of fractional order arch MEMS resonator according to claim 1, special
Sign is: the step S2 specifically:
Chebyshev multinomial is with the formal character of two recurrence formula
Ti+1(X)=2XTi(X)-Ti-1(X),T0(X)=1 (14)
Wherein X ∈ R and T1(X) it is defined as X, 2X, 2X-1 or 2X+1；
For [x1,…,xm]T∈Rm, a kind of polynomial strengthening form of Chebyshev is built as
ξ (X)=[1, T1(x1),…,Tn(x1),…,T1(xm),…,Tn(xm)] (15)
Wherein Ti(xj), i=1 ..., n, j=1 ..., m indicate that Chebyshev multinomial, ξ (X) indicate Chebyshev multinomial
Basis function vector, n indicate order；
For compacting upper any given unknown continuous function f (X),General based on Chebyshev neural network approaches
Theory adequately accurately approaches it, has
Wherein φ (t) is smooth weight vector；
There are Chebyshev neural networks
Wherein ε (X) > 0 is approximate error, ΩφAnd DXRespectively indicate the appropriate sets with compact border of φ (t) He X；If optimized parameter φ*
It is equal toWherein φ*Referred to as artificial amount；WhenShi You
To promote to take such as down conversion quickly in line computation and reduce Chebyshev neural network weight vector number
WhereinRelational expressionDeposit it is vertical,It is λi(t) estimated value, biIt is one small
Normal number；
By Young inequality, a mathematic(al) manipulation related with weight vector number of Chebyshev neural network is exported；
Step 1: defining first intermediate variable
Wherein tracking error e1(t) it is defined as e1(t)=x1(t)-xd(t), σ1Indicate positive design parameter；If Z1(t) → 0,So e1(t) → 0 He
Selection second has tracking errorIntermediate variable e2(t)=x2(t)-α2(t), wherein
σ2> 0 is a design parameter, α2(t) virtual controlling is indicated；Z is derived in the definition of Caputo fractional calculus1(t)
Derivative
Virtual controlling is selected as
Wherein k1> 0 represents control gain；
Based on lemma 1, following cline frequency distributed model is obtained:
Consider Liapunov stability criterion
Wherein
To V1(t) seeking time derivative
Step 2: selection fractional order Liapunov stability criterion
Wherein h2> 0,
To Z2(t) differential obtains
Wherein
f2() is a high-order nonlinear function, wherein such as h, β, γ and b11System parameter cannot accurately measure, and by
To the influences of internal and external factors, to establish accurate system model extremely difficult；Different external drives is to arch MEMS resonator meeting
Harmful oscillation is generated, it is this to vibrate the performance that reduce system to a certain extent；To solve these problems, it uses
Chebyshev neural network
In fact, cannot directly be found out due to calculating complexityTo solve this problem, design is based on hyperbolic sine letter
Several fractional order Nonlinear Tracking Differentiator estimates virtual controlling α2(t) Fractional Derivative
The wherein Nonlinear Tracking Differentiator state z based on hyperbolic sine function2,2WithIt is equal, r2> 0, ci> 0, i=1,2 and
di> 0, i=1,2 is design constant, exist be with T positive number relational expression
(27) and (28) are substituted into (26), are obtained
It is easy exportUsing lemma 1, cline frequency distributed model is further derived
Take the time-derivative of (25)
Wherein
It is inputted using being controlled below Nussbaum construction of function
Wherein k21> 0 and k22> 0 is control gain, has more new law
Wherein g2It is positive number；
More new law and control law substitute into (31), acquire
Theorem 1: assuming that 1 deposit vertical under conditions of, the sublevel arch MEMS resonator with unknown drive characteristic is considered, if institute
Antihunt self-adaptation control method (32) intervention being made of adaptive rate (33) and (34) is proposed, then all internal signals are kept
Bounded, while completely eliminating comprising the oscillation including chaotic behavior and asymmetric dead zone；
It proves: defining entire Liapunov candidate functions
By?
Wherein
DefinitionAbove formula is reduced to
Above formula both sides are multiplied simultaneouslyIt obtains
DefinitionAbove formula is integrated
Z1(t), Z2(t) andBelong to and compacts
Therefore, all signals in closed-loop system are all bounded；Further prove
So far, the proof of theorem 1 is completed.
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| CN113064347A (en) * | 2021-03-15 | 2021-07-02 | 贵州大学 | PMSM chaotic system self-adaptive control method considering asymmetric input and output constraints |
| CN113625573A (en) * | 2021-09-18 | 2021-11-09 | 金陵科技学院 | Fractional order system backstepping sliding mode control method influenced by asymmetric dead zone input |
| CN113625573B (en) * | 2021-09-18 | 2023-06-30 | 金陵科技学院 | Fractional order system backstepping sliding mode control method influenced by asymmetric dead zone input |
| CN114932560A (en) * | 2022-07-25 | 2022-08-23 | 山东贞元汽车车轮有限公司 | Automatic control method and system for grabbing hub by mechanical arm |
| CN114932560B (en) * | 2022-07-25 | 2022-11-22 | 山东贞元汽车车轮有限公司 | Automatic control method and system for grabbing hub by mechanical arm |
| CN115963868A (en) * | 2023-01-06 | 2023-04-14 | 贵州大学 | Acceleration preset performance control method for weak coupling fractional order MEMS resonator with event trigger mechanism |
| CN116720554A (en) * | 2023-08-11 | 2023-09-08 | 南京师范大学 | Method for realizing multi-section linear fitting neuron circuit based on FPGA technology |
| CN116720554B (en) * | 2023-08-11 | 2023-11-14 | 南京师范大学 | Method for realizing multi-section linear fitting neuron circuit based on FPGA technology |
| CN118607603A (en) * | 2024-07-25 | 2024-09-06 | 南京师范大学 | Method for realizing multi-scroll neuron circuit based on optimal linear approximation FPGA technology |
| CN118607603B (en) * | 2024-07-25 | 2024-10-18 | 南京师范大学 | Method for realizing multi-scroll neuron circuit based on optimal linear approximation FPGA technology |
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| NL2024372B1 (en) | 2020-11-30 |
| CN109613826B (en) | 2021-07-27 |
| NL2024372A (en) | 2020-08-13 |
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