CN106647277A - Self-adaptive dynamic surface control method of arc-shaped mini-sized electromechanical chaotic system - Google Patents

Abstract

The invention discloses a self-adaptive dynamic surface control method of an arc-shaped mini-sized electromechanical chaotic system. The self-adaptive dynamic surface control method comprises steps that an arc-shaped mini-sized electromechanical chaotic system dynamic model based on an Euler-Bernoull beam is established, and a system input output constraint condition is determined; virtual control input is used to form input of a first order filter by a Lyapunov function and a Levant tracker; an expansion state observer is designed to estimate a system state variable, and by combining with the first order filter, a Chebyshev neural network, a backstepping, and a Nussbaum function, actual control input is formed, and an output signal of a controller is input in the arc-shaped mini-sized electromechanical chaotic system; various link parameters in the controller are adjusted, and the adjustment of the parameters is completed. The influences of the arc-shaped mini-sized electromechanical chaotic system on closed loop control performance caused by the uncertainty factors, the non-measurable input output constraint, and the non-measurable state variable of the arc-shaped mini-sized electromechanical chaotic system are reduced.

Description

The self adaptation dynamic surface control method of arc microelectromechanicpositioning chaos system

Technical field

The present invention relates to arc microelectromechanicpositioning chaos system, and in particular to the self adaptation of arc microelectromechanicpositioning chaos system is moved State face control method.

Background technology

Micro Electro Mechanical System grows up on the basis of miniature electronic technology (semiconductor fabrication), has merged light The high-tech electronic of the fabrication techniques such as quarter, the processing of burn into film, LIGA, micro-silicon, non-silicon micro machining and precision optical machinery processing Mechanical devices.Micro Electro Mechanical System be collection microsensor, micro actuator, MIniature machinery structure, micro power micro power source, Signal transacting and control circuit, high-performance electronic integrated device, interface, communication etc. are in the microdevice or system of one.It is miniature Mechatronic Systems is a revolutionary new technology, is widely used in new high-tech industry, is that a n-th-trem relation n is sent out to the science and technology of country Exhibition, the key technology of of prosperous economy and national defense safety.Micro Electro Mechanical System lays particular emphasis on ultraprecise machining, is related to micro electric Son, material, mechanics, chemistry, mechanics subjects field.Its subject face cover power under miniature yardstick, electricity, light, magnetic, The physics such as sound, surface, chemistry, mechanistic each branch.

Arc microelectromechanicpositioning chaos system has sensitiveness to the primary condition under external environment condition, can be presented what is enriched very much Irregular chaotic oscillation is also easy to produce in dynamic behaviour, the i.e. course of work.Chaotic behavior greatly affects arc microelectromechanicpositioning to mix The stability and security of ignorant system, it is necessary to take steps to improve the performance of arc microelectromechanicpositioning chaos system.

Arc microelectromechanicpositioning chaos system phasor and sequential chart under different excitation amplitudes R is as shown in Figure 3 and Figure 4, very aobvious So, there is chaotic oscillation in system.Arc microelectromechanicpositioning chaos system Poincare Section is as shown in Figure 5.In Fig. 5 (a), (c), There are some points fixed in (e) and (f).When R reduces, constant track is attracted to start expansion, such as Fig. 5 (i) and (h) institute Show.When R further reduces, track starts deformation, shown in such as Fig. 5 (g) and (d).When R is equal to 0.02, system motion occurs Shown in unstable and chaos state, such as Fig. 5 (b).

Arc microelectromechanicpositioning chaos system periodic vibration state and chaotic motion, x are analyzed using bifurcation graphs₁Relative to sharp The bifurcation graphs for encouraging amplitude R are as shown in Figure 6.The system being clear that within the specific limits from regular motion to chaotic motion Dynamic behaviour, i.e. chaotic oscillation occur in region B, D, F and I, and four cycles were moved out on present region H and two cycles motion occurs On the G of region.

Meanwhile, in actual applications, due to factors such as actuator physical limit, component aging and external environment impacts So that actuator input and output are presented fan_shaped sedimentary body feature.This feature is inevitably present in actual control system In, and the hydraulic performance decline of closed-loop control system can be caused, even result in system unstable.Based on security reason and environmental protection Etc. aspect consideration, the state constraint and output constraint in control system be very important.Research discovery, arc microelectromechanicpositioning chaos Nonlinear compensation problem in system is related to the observation problem of position signalling and its higher derivative, and observer can be in system mode Not exclusively measurable situation realizes the closed-loop control of system, solves the problems, such as system mode On-line Estimation.

Micro Electro Mechanical System has the features such as nonlinearity, unknown parameters, chaotic oscillation and multivariable.Current micro computer Electric system research major part lays particular emphasis on dynamic analysis and the manufacturing, seldom solves its chaos up from self-adaptation control method Control problem.In addition, general control method does not account for the feature and feature of arc microelectromechanicpositioning chaos system, fail effectively There is fan_shaped sedimentary body, chaotic oscillation, Unknown control direction, difficult measuring state and state constraint feature etc. in solution system Control problem.Therefore, for the adaptive control problem of arc microelectromechanicpositioning chaos system, urgently need to propose that some are effective Control method, so as to reduce the adverse effect that various factors is caused to system, improve its performance, improve reliability and safety Property.

The content of the invention

Goal of the invention：In order to solve the problems, such as prior art, arc microelectromechanicpositioning chaos system is solved distributed There is fan_shaped sedimentary body, chaotic oscillation, Unknown control direction, difficult measuring state and state constraint feature etc. under static excitation Adaptive control problem, the present invention provides a kind of self adaptation dynamic surface control method of arc microelectromechanicpositioning chaos system.

Technical scheme：A kind of self adaptation dynamic surface control method of arc microelectromechanicpositioning chaos system, comprises the following steps：

(1) set up based on the arc microelectromechanicpositioning chaos system kinetic model of Euler-Bernoull beams, determine system Output constraint condition and fan_shaped sedimentary body condition；The non-scalar of arc microelectromechanicpositioning chaos system is listed according to kinetic model Equation, definition status variable；

(2) controller is designed, it is with ideal signal and then defeated by the output signal for comparing arc microelectromechanicpositioning chaos system Go out tracking error e₁；Design obstacle Lyapunov function, the obstacle Lyapunov function is used to ensure that output signal expires The output constraint condition of pedal system, error e₁Constitute with reference to Levant differential trackers Jing after the process of obstacle Lyapunov function Virtual controlling is input into, and virtual controlling input obtains filter output signal α Jing after firstorder filter filtering_2f；State variable is passed through Extended mode observer obtains variableBy filter output signal α_2fWith variableBy comparing further output errorIt is right Filter output signal α_2fDerivation obtains filter output signal derivative

(3) error e obtained according to step (2)₁WithFilter output signal derivativeBuild Nussbaum functions； Adaptive control laws are built in the framework of backstepping, according to state variable x₁And x₂Build and carry adaptive control laws Chebyshev neutral nets；Nussbaum functions and Chebyshev coupling of neural network are obtained into actual control input, institute State actual control input and arc microelectromechanicpositioning chaos system is input under the conditions of fan_shaped sedimentary body is met；

(4) Levant differential trackers, firstorder filter, extended mode observer, Chebyshev in controller are adjusted refreshing The parameter of Jing networks, detecting and tracking error e₁With the size that controller exports u；One error threshold of setting, when tracking error e₁It is little When error threshold, and during output signal value meet the constraint condition, complete the regulation of parameter.

Beneficial effect：Compare prior art, a kind of self adaptation of arc microelectromechanicpositioning chaos system that the present invention is provided Dynamic surface control method, with distributed static excitation have fan_shaped sedimentary body, chaotic oscillation, Unknown control direction, The arc microelectromechanicpositioning such as difficult measuring state and state constraint feature chaos system is object, based on Euler-Bernoull beam constructions Kinetic model, designs corresponding obstacle Lyapunov function ensureing output signal and strictly meets output constraint condition, ties The advantage that Levant differential trackers estimate preferable differential signal is closed, using Chebyshev neutral nets with arbitrarily small error The characteristic and extended mode observer of Nonlinear Function Approximation carrys out the immesurable state variable of online Prediction, reduces and physics is passed The restriction of sensor, eliminate the requirement to system mathematical models and accurate parameter, it is to avoid in traditional backstepping To coefficient expansion problem caused by virtual controlling repeatedly derivation, Unknown control direction problem is processed using Nussbaum functions, Self adaptation dynamic surface control device is constructed in the framework of backstepping.Present invention achieves ensureing system transients and steady-state behaviour Self Adaptive Control, relax to assumed condition known to system total state, arc microelectromechanicpositioning chaos system can be reduced not Impact of the certainty factor to closed-loop control performance.

Description of the drawings

Fig. 1 is the block diagram of the self adaptation dynamic surface control method of arc microelectromechanicpositioning chaos system；

Fig. 2 is arc microelectromechanicpositioning chaos system schematic diagram；

Fig. 3 is the arc microelectromechanicpositioning chaos system phasor under different excitation amplitudes R；

Fig. 4 is the arc microelectromechanicpositioning chaos system sequential chart under different excitation amplitudes R；

Fig. 5 is arc microelectromechanicpositioning chaos system Poincare Section；

Fig. 6 is the bifurcation graphs of arc microelectromechanicpositioning chaos system periodic vibration state and chaotic motion；

Fig. 7 is system fan_shaped sedimentary body schematic diagram；

Fig. 8 is system output constraint condition schematic diagram；

Fig. 9 isWithBetween differential tracker performance；

Figure 10 is x₁WithBetween observer performance；

Figure 11 is x₂WithBetween observer performance；

Figure 12 is the tracking performance under different excitation amplitudes R；

Figure 13 is the Nussbaum functions under different excitation amplitudes R；

Figure 14 is the actual control input under different excitation amplitudes R.

Specific embodiment

With reference to the accompanying drawings and detailed description, the present invention will be further described.

As shown in figure 1, the self adaptation dynamic surface control method of arc microelectromechanicpositioning chaos system, comprises the following steps：

(1) set up based on the arc microelectromechanicpositioning chaos system kinetic model of Euler-Bernoull beams, determine system Output constraint condition and fan_shaped sedimentary body condition；The non-scalar of arc microelectromechanicpositioning chaos system is listed according to kinetic model Equation, definition status variable；

(2) controller is designed, it is with ideal signal and then defeated by the output signal for comparing arc microelectromechanicpositioning chaos system Go out tracking error e₁；Design obstacle Lyapunov function, the obstacle Lyapunov function is used to ensure that output signal expires The output constraint condition of pedal system, error e₁Constitute with reference to Levant differential trackers Jing after the process of obstacle Lyapunov function Virtual controlling is input into, and virtual controlling input obtains filter output signal α Jing after firstorder filter filtering_2f；State variable is passed through Extended mode observer obtains variableBy filter output signal α_2fWith variableBy comparing further output errorIt is right Filter output signal α_2fDerivation obtains filter output signal derivative

(3) error e obtained according to step (2)₁WithFilter output signal derivativeBuild Nussbaum functions； Simultaneously according to state variable x₁And x₂Build the Chebyshev neutral nets with adaptive control laws；By Nussbaum functions with Chebyshev coupling of neural network obtains actual control input, and the actual control input is meeting fan_shaped sedimentary body bar Arc microelectromechanicpositioning chaos system is input under part；

(4) Levant differential trackers, firstorder filter, extended mode observer, Chebyshev in controller are adjusted refreshing The parameter of Jing networks, detecting and tracking error e 1 and controller export the size of u；One error threshold of setting, when tracking error e1 During less than error threshold, and during output signal value meet the constraint condition, complete the regulation of parameter.

In step (1), it is based on the arc microelectromechanicpositioning chaos system kinetics equation of Euler-Bernoull beams：

L is thin beam length in formula, and A is cross-sectional area, and b is thin beam width, C_vFor viscous damping system, d is thin cantilever thickness,For Young's modulus, I_yFor the moment of inertia, ρ is density, Ω₀For Simple Harmonic Load frequency, ε_a0For permittivity of vacuum, V_DCFor direct current Pressure, V_ACFor alternating voltage；

The edge-restraint condition of arc microelectromechanicpositioning chaos system is：

Assume that harmonic model is less than DC static load, by coordinate transform, using Galerkin decomposition methods, according to arc Microelectromechanicpositioning chaos system kinetics equation lists the non-scalar equation of arc microelectromechanicpositioning chaos system：

System variable is defined as in formula：

H=h₀/g₀；

In formula

Definition status variable x1, x2,

x₁=q,

Consider fan_shaped sedimentary body feature, and variable is substituted in the non-scalar equation of arc microelectromechanicpositioning chaos system ：

Wherein N (u) represents fan_shaped sedimentary body, and its characteristic relation is expressed as formula (1.6)；Y represents system output signal； s_l1＞ 0 and s_l2＞ 0 is oblique line l₁And l₂Slope, oblique line l₁And l₂For two fan-shaped borders, s_M=max (s_l1,s_l2).System Output y is required to meet certain constraints, and output constraint condition is | y |≤k_c1, wherein, k_c1Represent the threshold value of setting.

Self adaptation dynamic surface control device is described in detail below：

(1), Chebyshev nerve network systems

Chebyshev neutral nets are made up of a series of orthogonal polynomial, with binomial recurrence formula：

T_i+1(X)=2XT_i(X)-T_i-1(X),T₀(X)=1 (1.7)

X ∈ R and T in formula₁(X) X, 2X, 2X-1 or 2X+1 are normally defined.

Chebyshev neutral nets have approaches any non-linear company in a compact set with arbitrarily small trueness error The ability of continuous function.Compared with multilayer neural network, Chebyshev neutral nets have less amount of calculation.Chebyshev is more Item formula enhancement mode X=[x₁,…,x_m]^T∈R^mCan be defined as：

ξ (X)=[1, T₁(x₁),…,T_n(x₁),…,T₁(x_m),…,T_n(x_m)] (1.8)

T in formula_i(x_j), i=1 ..., n, j=1 ..., m represents Chebyshev multinomials, and n represents Chebyshev multinomials Divisor, ξ (X) represent Chebyshev polynomial basis functions.

Because Chebyshev neutral nets have omnipotent approximation capability, unknown nonlinear function f_CNN(X) can estimate：

f_CNN(X)=θ^*Tξ(X)+δ (1.9)

δ represents the neutral net approximate error of bounded in formula, and θ * represent best initial weights vector, and meet

Ω in formula_θAnd D_XRepresent boundary θ and X of compact set.In addition, to any positive definite constant δ₀, | δ |≤δ₀Meet.

(2), controller design

A) extended mode observer is designed

Using extended mode observer observational variable

Z in formula₀Represent reference signal f_r(τ) estimate, z_i, i=1 ..., n represents its derivative and k_i, i=1 ..., n tables Show design constant.

Knowable to above, if n is equal to 3, state observer extended below is designed

Observer errorMeet

In formulaWithRepresent x_i, i=1,2,3 estimate.k_o、k₁、k₂For design constant.

B) Bounded Errors variable is defined

In formulaRepresent x₂Estimate, indirect virtual controlling α_2fTo be given in ensuing content.

Continuous function N (η)：R → R is known as a Nussbaum gain function, has the property that：

Nussbaum gain functions are defined as：If V (.) and η (.) it is interval [0 ∞) be Smooth function, while V (t) >=0,So N (.) is smooth Nussbaum gain functions.Now have：

Constant c in formula₁＞ 0, g (t) are non-zero constants, c₀Represent certain rational constant, at the same V (t), η (t) andIt is interval [0 ∞) on must bounded.

Variable x₂Affected by physical sensors and cross-couplings minimization, generally can not be surveyed.Ask to solve this Topic, accurately estimates that its value, i.e. observation error are using extended mode observer

Select obstacle Lyapunov function

Whereind_bThe higher limit of ideal signal is represented,And restriction relationDo not disobey Instead.

To V₁Derivation

Y in formula₂Represent first-order filtering error and equal to α_2f-α₂。

Based on Levant differential trackers,Can substituteThen

Wherein r₁₁With r₁₂Represent constant, Represent the threshold value of setting；For Levant differential trackers Variable,For the output valve of Levant differential trackers.

With reference to obstacle Lyapunov function and Levant differential trackers, virtual controlling input is constituted：

C in formula₁Represent constant；

(1.21) are substituted into (1.19), then

C) adaptive control laws are built in the framework of backstepping, according to state variable x₁And x₂Build with certainly The Chebyshev neutral nets of suitable solution rule, the specific design method of its adaptive control laws is：

If firstorder filter time constant filter is τ₂, virtual controlling input is α₂, can obtain

Exist

To y₂Derivation is obtained

In formulaFor continuous function,

According to Young ' s inequality, inequality is obtained：

And then obtain errorFormula be：

WhereinFor observation errorDerivative, observation errorSo

f₂() is complicated nonlinear terms, α₂Derivative can cause system expand and computation burden.Due to foozle, The impact of environmental perturbation and modeling error, systematic parameter such as μ, h, α₁,β,ω₀,R,b₁₁It is uncertain.System parameter disturbance The chaotic oscillation of arc microelectromechanicpositioning chaos system can be caused.Therefore, these nonlinear characteristics are considered during control design case And factor.In view of Chebyshev neutral nets have universal approximation property, it is utilized to estimate f₂(·).That is, exist

In formulaδ₂For neutral net approximate error.

Introduce variable

In formulaRepresent λ₂Estimate；

Select Lyapunov function

γ in formula₂Represent design constant.

To V₂Derivation is obtained

From (1.21)

Inequality is obtained with reference to formula (1.22), (1.28), (1.32), (1.31)：

A in formula₂Represent design constant；

Actual design of control law is：

Adaptive control laws are designed as

C in formula₂And m₂Represent design constant；And have

There is relation(1.34)-(1.36) are substituted into (1.33), is obtained

In formula

Selection parameter τ₂Meet relationUsing Young inequality, (1.37) are rewritten as

There is fan_shaped sedimentary body under distributed static excitation for arc microelectromechanicpositioning chaos system, chaos is shaken Swing, difficult measuring state and state constraint feature, fusion extended mode observer (1.21) and differential tracker (1.20) are to control In device, self adaptation dynamic surface control device of the design with adaptive control laws (1.35)-(1.36) and wave filter (1.23) is such as (1.34) shown in, then system all signals such as e₁,Globally uniformly bounded, and output constraint is kept not to be breached, Tracking error is rapidly converged near zero.

Using all closed signal globally uniformly boundeds of Lyapunov theoretical proof closed-loop systems：

To V derivations：

C in formula₀=min { 2 × c₁,2×(c₂-0.5),m₂The Hes of ＞ 0

(1.39) both sides are multiplied by simultaneouslyObtain

It is givenIt is rightBy integrating to (1.40), obtain

Therefore, e₁,WithBelong to compact set

From above, all signal boundeds of arc microelectromechanicpositioning chaos system.Particularly, there is inequality

Due to y (τ)=e₁+x_d,With | x_d|≤d_b, it is easy to obtainTherefore, carried Control program ensure that y meet the constraint conditions.

According to the arc microelectromechanicpositioning chaos system of (1.5) description, Chebyshev polynomial orders are 3, micro- according to arc The non-scalar equation of type electromechanics chaos system, designs as follows by Chebyshev polynomial basis functions：

Although RBF neural is approached device and can solve the problems, such as dynamic surface coefficient expansion and with arbitrarily essence as one kind The ability of degree Nonlinear Function Approximation, but need to know center and the weights of Gaussian bases in advance during using RBF neural, And Chebyshev neutral nets only only need to input variable information.

Regulation parameter c₁, c₂, k₀, k₁, k₂, r₁₁, r₁₂, γ₂, a₂And m₂Ensure that tracking error tends to infinitely small, and ensure that Closed-loop system meets certain output constraint condition, makes self adaptation dynamic surface control technology have the robust to system parameter disturbance The rejection ability of property and chaotic behavior.

Arc microelectromechanicpositioning chaos system is as shown in Fig. 2 parameter value is α₁=7.993, β=119.9883, h=0.3, R=0.02, μ=0.1 and ω₀=0.4706.In order to solve ODE, using fourth order Runge-Kutta Algorithm for Solving, while The time of integration is set to more than 2000.

Varying reference signal x when given_d=0.35sin (3 τ).Controlled output meet the constraint condition | y | ＜ k_c1=0.39, can To obtainFan_shaped sedimentary body can be expressed as N (u)=(0.75+0.25sin (u)) u, such as Fig. 7 institutes Show.According to the self adaptation dynamic surface control method of arc microelectromechanicpositioning chaos system, each parameter selects as follows：By firstorder filter Timeconstantτ₂0.01 is set to, the parameter of extended mode observer selects to be k₀=k₁=20, k₂=19.Controller gain It is c to select with parameter₁=c₂=10, γ₂=1, m₂=15, a₂=5.Second-order differential tracker parameters select to be r₁₁=r₁₂=6,Initial value is set as 0.01.

Design second-order differential tracker On-line EstimationFig. 9 is illustratedWithBetween performance, two lines essentially coincide and Relative error is less than 0.05.Figure 10-11 illustrates the superior function of high-order extended mode observer.Knowable to figure, extended mode Observer can accurate estimated state variable, while relaxing the restriction to physical sensors.Although arc microelectromechanicpositioning chaos system System has the features such as fan_shaped sedimentary body, state constraint and chaotic oscillation, and observer is still accurately estimated with the error of very little State variable.

In order to contrast conveniently, the performance curve under different parameters is placed in a figure.Figure 12 illustrates e1 in different parameters Under tracking performance (it is also seen that x₁Tracking performance).It is obvious that 4 kinds of operating modes (i.e. R=0.01,0.02,0.15,0.25) Actual path coincide substantially with reference locus, and the difference between reality and theoretical performance curves be ± 0.01.

In addition, using | y | ＜ k after obstacle Li Ya spectrum husband's functions_c1Constraints is ensured.With the phasor and Fig. 4 of Fig. 3 Sequential chart contrast, realize in very short time using suggested plans rear arc microelectromechanicpositioning chaos system it is in stable condition, while with The chaotic oscillation of correlation thoroughly suppressed.

Because fan_shaped sedimentary body and control input direction are unknown, if do not adopted an effective measure controller is will necessarily result in Failure.Figure 13 represents the arc microelectromechanicpositioning chaos system Nussbaum function curves of different R values.Although systematic parameter is subjected to External interference, 4 Nussbaum function curves are consistent substantially.

Fan_shaped sedimentary body is present in the control input of arc microelectromechanicpositioning chaos system.Its presence affects control Performance even results in controller and jitter occurs.Figure 14 discloses the control input under different R values.As seen from the figure, tremble existing As greatly weakened, while set controller has certain robustness.

Claims (10)

1. a kind of self adaptation dynamic surface control method of arc microelectromechanicpositioning chaos system, it is characterised in that comprise the following steps：

(1) set up based on the arc microelectromechanicpositioning chaos system kinetic model of Euler-Bernoull beams, determine that system is exported Constraints and fan_shaped sedimentary body condition；The non-scalar side of arc microelectromechanicpositioning chaos system is listed according to kinetic model Journey, definition status variable；

(2) design controller, by compare the output signal of arc microelectromechanicpositioning chaos system with ideal signal so that export with Track error e₁；Design obstacle Lyapunov function, the obstacle Lyapunov function is used to ensure that output signal meets system The output constraint condition of system, error e₁Levant differential trackers are combined Jing after the process of obstacle Lyapunov function to constitute virtually Control input, virtual controlling input obtains filter output signal α Jing after firstorder filter filtering_2f；State variable is through extension State observer obtains variableBy filter output signal α_2fWith variableBy comparing further output errorTo filtering Device output signal α_2fDerivation obtains filter output signal derivative

(3) error e obtained according to step (2)₁WithFilter output signal derivativeBuild Nussbaum functions； Adaptive control laws are built in the framework of backstepping, according to state variable x₁And x₂Build with adaptive control laws Chebyshev neutral nets；Nussbaum functions and Chebyshev coupling of neural network are obtained into actual control input, it is described Actual control input is input into arc microelectromechanicpositioning chaos system under the conditions of fan_shaped sedimentary body is met；

(4) Levant differential trackers, firstorder filter, extended mode observer, Chebyshev nerve nets in controller is adjusted The parameter of network, detecting and tracking error e₁With the size that controller exports u；One error threshold of setting, when tracking error e₁Less than by mistake During difference limen value, and during output signal value meet the constraint condition, complete the regulation of parameter.

2. the self adaptation dynamic surface control method of arc microelectromechanicpositioning chaos system according to claim 1, its feature exists In the arc microelectromechanicpositioning chaos system kinetics equation in the step (1) based on Euler-Bernoull beams is：

$\begin{array}{c}\mathrm{\ρ}A\frac{{\∂}^{2}w}{\∂t^2}-\frac{\tilde{E}A}{2L}\[\underset{0}{\overset{L}{\∫}}({\left(\frac{\∂w}{\∂x}\right)}^2+2\frac{\∂w}{\∂x}\frac{{\mathrm{dw}}_0}{dx})dx\]\left(\frac{{\∂}^{2}w}{\∂x^2}+\frac{d^2{w}_0}{{\mathrm{dx}}^2}\right)+C_v\frac{\∂w}{\∂t}+\tilde{E}{I}_y\frac{{\∂}^{4}w}{\∂x^4}\\ =\frac{{\mathrm{\ϵ}}_{a0}b{(V_{DC}+V_{AC}\cos ({\mathrm{\Ω}}_{0}t\left)\right)}^2}{2{\left(g_0-w_0-w\right)}^2}\end{array}$

L is thin beam length in formula, and A is cross-sectional area, and b is thin beam width, C_vFor viscous damping system, d is thin cantilever thickness,For Young's modulus, I_yFor the moment of inertia, ρ is density, Ω₀For Simple Harmonic Load frequency, ε_a0For permittivity of vacuum, V_DCFor DC voltage, V_AC For alternating voltage；

The edge-restraint condition of arc microelectromechanicpositioning chaos system is：

$w(0,t)=w(L,t)=0,\frac{\∂w}{\∂x}{|}_{x=0}=\frac{\∂w}{\∂x}{|}_{x=L}=0.$

3. the self adaptation dynamic surface control method of arc microelectromechanicpositioning chaos system according to claim 2, its feature exists In by coordinate transform, using Galerkin decomposition methods, according to arc microelectromechanicpositioning chaos system kinetics equation arc being listed The non-scalar equation of microelectromechanicpositioning chaos system：

$\stackrel{\·\·}{q}+\mathrm{\μ}\stackrel{\·}{q}+(1+2h^2{\mathrm{\α}}_1)q-\frac{\mathrm{\β}(1+2Rcos({\mathrm{\ω}}_0\mathrm{\τ}\left)\right)}{2b_{11}\sqrt{{(1+h-q)}^3}}-3{\mathrm{\α}}_1{\mathrm{hq}}^2+{\mathrm{\α}}_1{q}^3=0$

System variable is defined as in formula：

H=h₀/g₀；

In formula

Consider fan_shaped sedimentary body feature, definition status variable x₁=q, state variableAnd variable is substituted into into arc In the non-scalar equation of microelectromechanicpositioning chaos system：

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2,y=x_1\\ {\stackrel{\·}{x}}_2=-{\mathrm{\μx}}_2-(1+2h^2{\mathrm{\α}}_1)x_1+\frac{\mathrm{\β}(1+2Rcos({\mathrm{\ω}}_0\mathrm{\τ}\left)\right)}{2b_{11}\sqrt{{(1+h-x_1)}^3}}+3{\mathrm{\α}}_1{\mathrm{hx}}_1^2-{\mathrm{\α}}_1{x}_1^3+N\left(u\right)\end{array}\right.$

$\left\{\begin{array}{c}{s}_{l1}u\left(\mathrm{\τ}\right)\≤N\left(u\right)\≤s_{l2}u\left(\mathrm{\τ}\right),u\left(\mathrm{\τ}\right)\≥0\\ s_{l1}u\left(\mathrm{\τ}\right)\≥N\left(u\right)\≥s_{l2}u\left(\mathrm{\τ}\right),u\left(\mathrm{\τ}\right)\≤0\end{array}\right.$

Wherein N (u) represents fan_shaped sedimentary body, s_l1＞ 0 and s_l2＞ 0 is oblique line l₁And l₂Slope, oblique line l₁And l₂For sector Two borders；Y represents system output signal.

4. the self adaptation dynamic surface control method of arc microelectromechanicpositioning chaos system according to claim 3, its feature exists In in step (1), output constraint condition is | y |≤k_c1, wherein, k_c1Represent the threshold value of setting.

5. the self adaptation dynamic surface control method of arc microelectromechanicpositioning chaos system according to claim 4, its feature exists In in step (2), the barrier function is obstacle Lyapunov function, and formula is：

$V_1=\frac{1}{2}ln\frac{k_{b_1}^2}{k_{b_1}^2-e_1^2}$

Whereind_bRepresent the higher limit of ideal signal.

6. the self adaptation dynamic surface control method of arc microelectromechanicpositioning chaos system according to claim 5, its feature exists In in step (2), with reference to obstacle Lyapunov function and Levant differential trackers, composition virtual controlling input：

${\mathrm{\α}}_2=-(k_{b_1}^2-e_1^2)(c_1+\frac{1}{2})e_1+{\mathrm{\θ}}_{12}$

C in formula₁Represent constant；θ₁₂For the output valve of Levant differential trackers, calculated by below equation：

$\left\{\begin{array}{c}{\stackrel{\·}{\mathrm{\θ}}}_{11}=-r_{11}|{\mathrm{\θ}}_{11}-x_d{|}^{0.5}sign({\mathrm{\θ}}_{11}-x_d)+{\mathrm{\θ}}_{12}\\ {\stackrel{\·}{\mathrm{\θ}}}_{12}=-r_{12}sign({\mathrm{\θ}}_{12}-{\stackrel{\·}{\mathrm{\θ}}}_{11})\end{array}\right.$

Wherein r₁₁With r₁₂Represent constant, Represent the threshold value of setting；θ₁₁For the change of Levant differential trackers Amount.

7. according to the self adaptation dynamic surface control method of the arbitrary described arc microelectromechanicpositioning chaos system of claim 3 to 6, its It is characterised by, in step (2), the exponent number of extended mode observer is set to 3, calculates the estimate of state variable：

$\left\{\begin{array}{c}\stackrel{\·}{\hat{x}}=-k_0|{\hat{x}}_1-x_1{|}^{2/3}sign({\hat{x}}_1-x_1)+{\hat{x}}_2\\ {\stackrel{\·}{\hat{x}}}_2=-k_1|{\hat{x}}_2-{\stackrel{\·}{\hat{x}}}_1{|}^{1/2}sign({\hat{x}}_2-{\stackrel{\·}{\hat{x}}}_1)+{\hat{x}}_3+u\\ {\stackrel{\·}{\hat{x}}}_3=-k_{2}sign\left({\hat{x}}_3-{\stackrel{\·}{\hat{x}}}_2\right)\end{array}\right.$

$x_3=f_2(\·)=-(1+2h^2{\mathrm{\α}}_1)x_1+\frac{\mathrm{\β}(1+2Rcos({\mathrm{\ω}}_0\mathrm{\τ}\left)\right)}{2b_{11}\sqrt{{(1+h-x_1)}^3}}-{\mathrm{\μx}}_2-{\mathrm{\α}}_1{x}_1^3+3{\mathrm{\α}}_1{\mathrm{hx}}_1^2$

In formulaIt is x_i, i=1,2,3 estimate, k_o、k₁、k₂For design constant.

8. the self adaptation dynamic surface control method of the arc microelectromechanicpositioning chaos system according to claim 1 or 2 or 3, its It is characterised by, in step (3), Nussbaum gain functions are defined as：

$N\left(\mathrm{\η}\right)=e^{{\mathrm{\η}}^2}\×cos\left(\frac{\mathrm{\π}}{2}\mathrm{\η}\right)$

$\underset{s\→+\mathrm{\∞}}{\lim }\sup \frac{1}{s}{\∫}_0^{s}N\left(\mathrm{\η}\right)d\mathrm{\η}=+\mathrm{\∞}$

$\underset{s\→+\mathrm{\∞}}{\lim }inf\frac{1}{s}{\∫}_0^{s}N\left(\mathrm{\η}\right)d\mathrm{\η}=-\mathrm{\∞}$

If V (.) and η (.) it is interval [0 ∞) be smooth function, while V (t) >=0,So N (.) is smooth Nussbaum gain functions, now have：

$V\left(t\right)\≤c_0+e^{-c_{1}t}{\∫}_0^{t}g\left(\mathrm{\τ}\right)N\left(\mathrm{\η}\right)\stackrel{\·}{\mathrm{\η}}{e}^{c_1\mathrm{\τ}}d\mathrm{\τ}+e^{-c_1\mathrm{\τ}}{\∫}_0^t\stackrel{\·}{\mathrm{\η}}{e}^{c_1\mathrm{\τ}}d\mathrm{\τ}$

Constant c in formula₁＞ 0, g (t) are non-zero constants, c₀Represent certain rational constant, at the same V (t), η (t) andIt is interval [0 ∞) on must bounded.

9. the self adaptation dynamic surface control method of arc microelectromechanicpositioning chaos system according to claim 3, its feature exists In the method for building adaptive control laws in step (3) in the framework of backstepping is：

If firstorder filter time constant filter is τ₂, virtual controlling input is α₂, can obtain

${\mathrm{\τ}}_2{\stackrel{\·}{\mathrm{\α}}}_{2f}+{\mathrm{\α}}_{2f}={\mathrm{\α}}_2,{\mathrm{\α}}_{2f}\left(0\right)={\mathrm{\α}}_2\left(0\right)$

Exist

To y₂Derivation is obtained

In formulaFor continuous function,

According to Young ' s inequality, inequality is obtained：

$y_2{\stackrel{\·}{y}}_2\≤-\frac{y_2^2}{{\mathrm{\τ}}_2}+y_2^2+\frac{1}{4}{B}_2^2$

And then obtain errorFormula be：

WhereinFor observation errorDerivative, observation errorSo f₂() is nonlinear terms, is estimated by Chebyshev neutral nets；

ExistIn formulaδ₂For neutral net approximate error, variable is introduced Represent λ₂Estimate；

Select Lyapunov function

γ in formula₂Represent design constant；

To V₂Derivation

With reference tof₂(·)、Formula obtain inequality：

$\begin{array}{c}{\stackrel{\·}{V}}_2\≤{\hat{e}}_2\left(\frac{1}{2a_2^2}{\mathrm{\λ}}_2{\hat{e}}_2{\mathrm{\ξ}}_2^T{\mathrm{\ξ}}_2+\frac{1}{2}{\hat{e}}_2-{\stackrel{\·}{\mathrm{\α}}}_{2f}+\frac{e_1}{k_{b_1}^2-e_1^2}+s_{M}u\right)+\frac{1}{4}{B}_2^2-\left(c_1+\frac{1}{2}\right)e_1^2\\ +\frac{1}{2}{\mathrm{\δ}}_{20}^2+\frac{a_2^2}{2}+\left(l_{{\mathrm{\θ}}_2}+\mathrm{\ν}\right)\left|\frac{e_1}{k_{b_1}^2+e_1^2}\right|+\left|{\hat{e}}_2\right|\mathrm{\ν}+\left(1-\frac{1}{{\mathrm{\τ}}_2}+\frac{0.5}{|k_{b_1}^2-e_1^2|}\right)y_2^2\\ +\frac{1}{{\mathrm{\γ}}_2}\widetilde{\mathrm{\λ}}{\stackrel{\·}{\widehat{\mathrm{\λ}}}}_2\end{array}$

A in formula₂Represent design constant；

Actual design of control law is：

$u=N\left(\mathrm{\η}\right)\[(c_2+\frac{1}{2}){\hat{e}}_2+\frac{e_1}{k_{b_1}^2-e_1^2}+\frac{1}{2a_2^2}{\widehat{\mathrm{\λ}}}_2{\hat{e}}_2{\mathrm{\ξ}}_2^T{\mathrm{\ξ}}_2-{\stackrel{\·}{\mathrm{\α}}}_{2f}\]$

Adaptive control laws are designed as:

${\stackrel{\·}{\widehat{\mathrm{\λ}}}}_2=\frac{1}{2a_2^2}{\mathrm{\γ}}_2{\mathrm{\ξ}}_2^T{\mathrm{\ξ}}_2{\hat{e}}_2^2-m_2{\widehat{\mathrm{\λ}}}_2,{\widehat{\mathrm{\λ}}}_2\left(0\right)>0$

$\stackrel{\·}{\mathrm{\η}}=\left(\right(c_2+\frac{1}{2}){\hat{e}}_2+\frac{e_1}{k_{b_1}^2-e_1^2}+\frac{1}{2a_2^2}{\widehat{\mathrm{\λ}}}_2{\hat{e}}_2{\mathrm{\ξ}}_2^T{\mathrm{\ξ}}_2-{\stackrel{\·}{\mathrm{\α}}}_{2f}){\hat{e}}_2$

C in formula₂And m₂Represent design constant；And have

10. according to the self adaptation dynamic surface control method of the arbitrary described arc microelectromechanicpositioning chaos system of claim 3 to 6, Characterized in that, Chebyshev polynomial orders are 3, and according to the non-scalar equation of arc microelectromechanicpositioning chaos system, will The design of Chebyshev polynomial basis functions is as follows：

${\mathrm{\ξ}}_2(x_1,{\hat{x}}_2)=\[1,x_1,2x_1^2-1,4x_1^3-3x_1,{\hat{x}}_2,2{\hat{x}}_2^2-1,4{\hat{x}}_2^3-3{\hat{x}}_2\].$