AU2018100710A4 - Adaptive Neural Network Control Method For Arch-Shaped Microelectromechanical System - Google Patents

Adaptive Neural Network Control Method For Arch-Shaped Microelectromechanical System Download PDF

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AU2018100710A4
AU2018100710A4 AU2018100710A AU2018100710A AU2018100710A4 AU 2018100710 A4 AU2018100710 A4 AU 2018100710A4 AU 2018100710 A AU2018100710 A AU 2018100710A AU 2018100710 A AU2018100710 A AU 2018100710A AU 2018100710 A4 AU2018100710 A4 AU 2018100710A4
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Shaobo LI
Shaohua Luo
Trandinh PHUNG
Guanci YANG
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Guizhou University
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    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
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Abstract

The present disclosure discloses an adaptive neural network control method for an arch-shaped microelectromechanical system. The method includes: a step (a) of constructing a system model of an arch-shaped microelectromechanical system based on an Euler-Bernoulli beam; and a step (b) of constructing an adaptive neural network controller for suppressing chaotic oscillations of the arch-shaped microelectromechanical system and ensuring state constraints of the system; and when the controller is constructed, ensuring an unviolated output constraint of the arch-shaped microelectromechanical system using a symmetric barrier Lyapunov function, adopting a RBF neural network having an adaptive rule to estimate an unknown nonlinear function with an arbitrarily small error, introducing an extended state tracking differentiator to address a problem of repeated derivative calculations required for a virtual control item in a backstepping control, designing a state observer to obtain immeasurable state information, and fusing the extended state tracking differentiator and the state observer in a backstepping framework. The present disclosure is convenient for an analysis and proof of a stability, and has characteristics including a low requirement on modeling accuracy, a low calculation complexity, a quick arithmetic speed, a good operation stability of the system, and high motion accuracy. vaibl -~ variable X o Cd rl Cd4 , 4 -.--. oil Cd1 cn

Description

TECHNICAL FIELD [2] The present disclosure relates to an arch-shaped microelectromechanical system, and specifically to an adaptive neural network control method for an arch-shaped microelectromechanical system.
BACKGROUND [3] A microelectromechanical system (MEMS) is a complex micro-device or an independent intelligent system that integrates micro sensor, micro actuator, micro energy source, signal processing and control circuits, high-performance electronic integrated device, interfaces, communication, etc. The MEMS technology is a revolutionary new technology involving many subject areas such as microelectronics, information and automatic control, mechanics, material, and dynamics. The MEMS technology is widely used in new high technology industries and is a key technology related to the science and technology development, economic prosperity and security of a nation.
[4] Nonlinear factors (such as uncertainty in modeling, environmental changes, aging of components, and self-excited vibrations) are inevitable. Once a MEMS fails, it will certainly lead to chain reactions, and the failure of an entire equipment, thus causing major safety accidents, economic losses and
2018100710 25 May 2018 adverse social impacts. Due to the influences by factors such as manufacturing defects, external disturbances, hysteresis loops and immeasurable states, and the uncertainty in mathematical models of the MEMS, it is difficult to obtain an accurate model for a control object. An arch-shaped MEMS is a high-order, multi-field coupling and parameter time-variable complex nonlinear system. However, the existing control methods for MEMS cannot properly solve control problems of output constraints, chaotic oscillations, immeasurable states and unknown dynamics in the arch-shaped MEMS.
SUMMARY [5] The purpose of the present disclosure is to provide an adaptive neural network control method for an arch-shaped microelectromechanical system. The present disclosure facilitates the stability analysis and proof, and has characteristics including low requirement on modeling accuracy, low calculation complexity, high arithmetic speed, good operation system stability, and high motion accuracy.
[6] Technical solution provided by the present disclosure: An adaptive neural network control method for an arch-shaped microelectromechanical system includes the following steps.
[7] a) a system model of an arch-shaped microelectromechanical system is constructed based on an Euler-Bernoulli beam to obtain ^(t) + m^(t) + h+ 2.h2aAq(t) — ( 0 )) _τ,π (t) + a,q2 (t) + u(t) = 0 (]_)
20,,^(1 + //-^(/))3 [8] In expression (1), mr h, a, , b, R and bn represent dimensionless parameters, q(t) represents a state variable, w0 represents a frequency, and «(/) represents a control input.
2018100710 25 May 2018 [9] b) an adaptive neural network controller for suppressing chaotic oscillations of the arch-shaped microelectromechanical system and ensuring state constraints of the system is constructed. When the controller is constructed, an unviolated output constraint of the arch-shaped microelectromechanical system is ensured using a symmetric barrier Lyapunov function, an RBF neural network having an adaptive rule is adopted to estimate an unknown nonlinear function with an arbitrarily small error, an extended state tracking differentiator is introduced to address a problem of a derivative repeatedly required for a virtual control item in a backstepping control, a state observer is designed to obtain immeasurable state information, and the extended state tracking differentiator and the state observer are fused in a backstepping framework.
[10] In the adaptive neural network control method for an arch-shaped microelectromechanical system, if x, = q(t) and x2=$/) are defined as new variables, expression
Error! Reference source not found, of the system model in step a is rewritten as:
4(-=x2,y = x[ b (l + 2A cos (wot 2bn^(l + h-xf [11] In expression (2), the system output 1 satisfies a constraint condition, that is, |r|A r where kcl>0.
[12] In b) of the adaptive neural network control method for an arch-shaped microelectromechanical system, the process of constructing the adaptive neural network controller for suppressing the chaotic oscillations of the arch-shaped microelectromechanical system and ensuring the state constraints of the system includes the following steps.
[13] bl) the state observer is constructed and the expression x, +
)) + 3a, Ax,2 — a,x, + u (0 (2) (3) .
2018100710 25 May 2018 is :
|*7= £+ Kfy- fi) f$= G(i/)+ /,(1)+ K,(y- fi) [14] In expression (3), x= [fi,x2J represents an estimation value of x= [χ.χ;] . Through an appropriate selection for /< [/<./<,], „ 4 K, ly
B= ς 1 y e k ou e λ2 Ύ [15] Expression (4) belongs to a Hurwitz matrix, [16] If an error of the state observer and a nonlinear item are defined as z=x.x and z , \ Z?(l+ 2Rcos(wnt/\ , /,(x)= - (l+ 2h~a/)xx + -, - mx, - af+ 3a hx~
2b ^(1+ h- xf
Then,
N= Bz + f (x) (5:
[17] Herein, /(χ)= | /2(x)- /2(*)f and z - [z, z2J [18] A Lyapunov function is selected as
6) [19] In expression (6), p represents a positive definite matrix, and satisfies the relationship btp+pb=-i· [20] Using a Young's inequality, a derivative of h0 is obtained as :
- zrz + 4t||y>||2 (7) .
[21] In step b2, the adaptive neural network controller is constructed, and the process is as follows.
/44
2018100710 25 May 2018 [22]
An estimation value of a nonlinear item the system model is expressed by the following RBF neural network expression:
of [23] In expression (8), ¢ = ,¾ J ek represents a weight vector, />1 represents a node number of a neuron, x^R represents an input vector, and x(X) = [x, (X),x2 (V)-L ,x, (Y)]r eR‘ . xfiX) represents a Gaussian function expressed as:
x,. (Y) = exp (X —m )T (X —ml) i = 1,L ,/ [24] In expression (9) , s, represents a width of the Gaussian function, and mt = [m,,zw2,L .mf\ represents a weight factor.
[25] A first error function is defined as s[=x[-xr, where xr represents a reference trajectory. A symmetric barrier Lyapunov function is selected as:
[26] In expression (15), f=f-xu, and the constraint condition βι|<Α is not violated.
[27] Since the state variable x2 is immeasurable, the state observer is used to estimate it.
[28] A second error function is defined as s2^x2-a2r where a2 represents a virtual control.
[29] A derivative of fi is given as:
2018100710 25 May 2018 s, (s2 + α2 -Jt&+z2) + f&
(16) .
[30] In expression (16), -sj .
[31] The virtual control is obtained as a2=-k^sl+jSi, where ρ>0, thus obtaining:
< -ζ Ί z - c,si + -Et- +1 k, + 4i||p||H2 (17) .
[32] A derivative of - is calculated as: s2 $=fu„ (·)+«(/) (18) .
[33] The RBF neural network is used to approximate a nonlinear function: /„,() = f (t)x2 (ρ, x2) + e2 (x,, x2), where e2(x,,x2)>0 .
[34] A number of weight vectors for the RBF neural network are decreased by taking actions. Using the Young's inequality, the expression is obtained as:
qT2 (i)x2 (x,, x2) = — y 2 (t)xT2 (x,, x2 )x2 (x,, x2) + 2n (19) .
[35] In expression (19), y2 0)=||^ (/)¾ 0)||, «2>0, =y2(t}—yr and y2(t) represents an estimation value of y2(r).
[36] An estimation value of the derivative of a2 is obtained using expression (20) of the extended state tracking differentiator :
=v η -bjxfal{d7j} , \ (20) · 7 =-bj2fal{de,hej) [37] There is a nonlinear function:
fal(de,h^ = < de ‘ , where Zr,y' = l,L,«, and / = 1,2 b p sign(Ae/),k |><
2018100710 25 May 2018 < d, represent feedback gains, hej represents a tracking differentiator error, de >0 and ae >0.
[38] A Lyapunov function is selected as:
[39] In expression (21), g2>0 .
(21) [40] An extended state tracking differential item is used v 22 to estimate and a time derivative of y is given as:
= A (Λ (·) + M 0) ” v 22) + —/°2 (f)i&2 (f) §2 (22) [41] It can be known according to expression (20) that he2=v2l-a2. There exists the inequality \v 22-c&\<hx.
[42] Expressions (18) and (20) are substituted into expression (17) to obtain:
!&<-zrz -c,q2 +s2^j-Ty2(t)xT2 (x, ,x2 )x2 (x, ,x2 ).?2 +kf2 + E +u(t)-v 22 +|,;|/+4£||p||H2 +—7°2(Φ&2(0+γ+ζ
Si 2 (23) .
[43] In expression (23), |<?2(χ,,χ2)|<<?2 .
[44] The actual control input m(/) and its adaptive rule y&(/) are respectively obtained as:
M (/) = -[ C21 +|y2 2 OX (^α)α(αΛ)α +v 22 -c22signs2 (24 ) , (25) .
2018100710 25 May 2018 and
A (0 = (X1 ’ A )-L (a , f )¾ - m 2y 2 (i) ln2 [45] In expression (24) or (25), c21>0, c22>0, and m2>0 .
[46] Beneficial effects: As compared with the existing technologies, the present disclosure has the following advantages. In the adaptive neural network control method for the arch-shaped microelectromechanical system of the present disclosure, a symmetric barrier Lyapunov function is used to ensure that an output constraint of the arch-shaped microelectromechanical system is not violated, and meanwhile, it is convenient for an analysis and proof of a stability; a RBF neural network having an adaptive rule is adopted to estimate an unknown nonlinear function with an arbitrarily small error, which lowers requirements of accurately constructing the system model and suppresses impacts from system parameter variations; and a problem that a derivative is repeatedly required for a virtual control item in a traditional backstepping control is addressed by introducing an extended state tracking differentiator, which reduces the calculation complexity and accelerates the arithmetic speed. In the control method of the present disclosure, the extended state tracking differentiator and a state observer are fused in a backstepping framework. According to the method, requirements for measurable state variables and relevant physical sensors are reduced, impacts from the chaotic oscillations and the output constraints on the system are suppressed, and the operation stability and motion accuracy of the system are improved.
BRIEF DESCRIPTION OF THE DRAWINGS [47] Fig. 1 is a schematic diagram of an arch-shaped
2018100710 25 May 2018 microelectromechanical system;
[48] Fig. 2 illustrates phase diagrams under different R values;
[49] Fig. 3 illustrates time histories under different R values;
[50] Fig. 4 illustrates a largest Lyapunov exponent;
[51] Fig. 5 is a bifurcation diagram for R - ;
[52] Fig. 6 illustrates tracking performances under different R values;
[53] Fig. 7 illustrates an observer performance between and ή ;
[54] Fig. 8 illustrates an observer performance between x2 and x2 ; and [55] Fig. 9 illustrates control inputs under different R values .
DETAILED DESCRIPTION OF EMBODIMENTS [56] The present disclosure is further described below in combination with the accompanying drawings and the embodiment. However, the accompanying drawings and the embodiment are not the bases for limiting the present disclosure.
[57] Embodiment [58] An adaptive neural network control method for an arch-shaped microelectromechanical system includes the following steps.
[59] In step a, in order to reveal inherent properties of the arch-shaped microelectromechanical system and easy to design
2018100710 25 May 2018 a controller, the nonlinear dynamics of the arch-shaped microelectromechanical system is investigated using phase diagrams, time histories, a largest Lyapunov exponent and a bifurcation diagram, and a system model of the arch-shaped microelectromechanical system is constructed based on an Euler-Bernoulli beam, to obtain:
tfy)+mt^t) + {l + 2h2a(}q(t) — ( θ )) _(tj+ayf (t) + u(t) = 0 ( q )
20,,^(1 + /:-7(7))3 [60] In expression (1), m, h, a, , b, R, and bn represents dimensionless parameters, q(t) represents a state variable, w0 represents a frequency, and u(t) represents a control input. The schematic diagram of the arch-shaped microelectromechanical system is shown in Fig. 1.
[61] In step b, an adaptive neural network controller for suppressing chaotic oscillations of the arch-shaped microelectromechanical system and ensuring state constraints of the system is constructed. When the controller is constructed, an unviolated output constraint of the arch-shaped microelectromechanical system is ensured using a symmetric barrier Lyapunov function, a RBF neural network having an adaptive rule is adopted to estimate an unknown nonlinear function with an arbitrarily small error, an extended state tracking differentiator is introduced to address a problem of a derivative repeatedly required for a virtual control item in a backstepping control, a state observer is designed to obtain immeasurable state information, and the extended state tracking differentiator and the state observer are fused in a backstepping framework.
[62] If x,=7(r) and x2=$t) are defined as new variables, expression Error! Reference source not found, of the system model in step a is rewritten as:
(2) .
2018100710 25 May 2018 , , b (l + 2Acos(w0i))
- — rwc2 — il + 2/z a} JXj H-- —axxx + u(t)
Ib/Q + h-x/ [63] In expression (2), the system output y satisfies a constraint condition, that is, |y|<iri, where kcl >0 .
[64] For the convenience of designing the controller, operation mechanisms of the arch-shaped microelectromechanical system need to be revealed. Parameters of the system are set to be 6/,=7.993, 6=119.9883 , 6 = 0.3, 7? = 0.02, /«=0.1, w0 =0.4706 and w(i)=0 . The differential equation is solved using the fourth-order Runge-Kutta algorithm. Figs. 2-3 illustrate phase diagrams and time histories under different excitation amplitudes R . It can be known from Figs. 2-3 that chaotic oscillations appear in the arch-shaped microelectromechanical system.
[65] As illustrated in Fig. 4, the Lyapunov exponent becomes a positive value in a short time. It is obvious that chaotic oscillations appear in the arch-shaped microelectromechanical system. Fig. 5 is a bifurcation diagram for R -x, of the arch-shaped microelectromechanical system, and further reveals periodic oscillation states and chaos motions.
[66] The chaotic oscillations of the arch-shaped microelectromechanical system inevitably lead to the deterioration of the system performance. Therefore, it is necessary to provide an effective control scheme to suppress the chaotic oscillations of the arch-shaped microelectromechanical system.
[67] In step b, the process of constructing the adaptive neural network controller for suppressing the chaotic oscillations of the arch-shaped microelectromechanical system includes the following steps.
2018100710 25 May 2018 [68] In step bl, the state observer is constructed, and the expression is:
f-p= Ϊ+ K (> - ') |A= G(i/)+ /,(1)+ Kjy- /) ο:
[69] In expression (3), x = [x,,x2J represents an estimation value of x= [x,,x,J . Through an appropriate selection for /< [/<./<,], „ 4 K, iv
5= S u e k 0u e vu [70] Expression (4) belongs to the Hurwitz matrix.
[71] If an error of the state observer and a nonlinear item are defined as z=x- I and z , \ Z?(l+ 2Rcos(wntI\ , /,(x)= -(1+ 2h~a[)xx+ -. - mx , - af+ 3a hx~ 2Z,nW+ h~ X/
N= Bz + f (x) :5) [72] Herein, /(x)= |) /2(x)-/2(*)f and z = [z, zj [73] A Lyapunov function is selected as
V, = zPz [74] In expression (6), P represents a positive definite matrix, and satisfies the relationship btp+pb=-i.
[75] Using the Young's inequality, a derivative of h0 is obtained as :
(7) .
[76] In step b2, the adaptive neural network controller is constructed, and the process is as follows.
2018100710 25 May 2018 [77] An estimation value of a nonlinear item frhJ(x) of the system model is expressed by the following RBF neural network expression:
[78] In expression (8), ¢ = ,<?2,L ,¾ J <=Rl represents a weight vector, />1 represents a node number of a neuron, x-R represents an input vector, and x(Y) = [x, (Y),x2 (Y),L ,x; (Y)]r e/?'. x,(Y) represents a Gaussian function expressed as:
x. (X) = exp
2s:
, and z = l,L ,/ [79] In expression (9) , si represents a width of the Gaussian function, and mi = ,min]T represents a weight factor.
[80] It can be known according to the neural network expression (8) that the following expression is obtained:
SUP \frbr (Χ)-9Γ 0MX)| ~e
XeDx 1 1 (10).
[81] In expression (10), e>0, ty, and Dx represent compact sets of q(t) and X . An ideal parameter f is defined to be equal t Ο arg min sup\frhf (x)-frhf (x,q)\ , and there exists f{t) = q(t\-q (/) at the
XeDj 1 same time.
[82] Assumption 1: A reference trajectory xr is bounded (i.e., |χ,.|<χ„, where x„>0), and A is also bounded.
[83] Assumption 2 : A constant L>0 also exists and satisfies :
|f (x)- f (x)| £ £(|χ, - x,| + L + |x,. - x,|),z = 1,L ,n (11) .
[84] In expression (11), Jftx)= 7(χ)- 7(χ), and f(x) represents an
2018100710 25 May 2018 estimation value of f,(x) .
[85] Lemma 1: For kh >0,/ = 1,L ,n , Z:= {se j :|y| <kh ,i = 1,L cz j and
N:= j ' xZ —> j /+” are defined as open sets.
[86] In consideration of the system, }&=he(t,h) (12) [87] In expression (12), h :=[w. .s]7 eN , and A,: j + xN -> j /+ is piecewise continuous in t and locally Lipschitz in Z . For U: j ' j + and F : y j ., there are v| S,.|-> kh , and n\ (H)<U(w)<ZM,(||w||) (13) [88] In expression (13), m, and tf represent class K„ functions .
[89] It is assumed that V(h) := Σ Vx (s,.) + U(w) and si e (~kh,kh} . If /-1 there exists an inequality t&= — h <0 dh 1 (14) , [90] A first error function is defined as sx=xx-xr, where % represents the reference trajectory. A symmetric barrier Lyapunov function is selected as:
V, = —ln
-+Vn kl -si ° (15) [91] In expression (15), ^,=^,,-+,,, and the constraint condition k|<U is not violated.
[92] Since a state variable %2 is immeasurable, the state
2018100710 25 May 2018 observer is used to estimate it.
[93] A second error function is defined as s2=x2-a2, where a2 represents a virtual control.
[94] A derivative of f is given as:
p s.(s2+a2-&+z2\ p
1-r-l2 + f& (16) .
[95] [96] g >0
In expression (16), -s( .
The virtual control is obtained as a2 =-ky^ + &-, where thus obtaining:
-c,s( + ^- + / k, + 4L (17) .
[97] A derivative of s2 is calculated as:
^ = Z„(j + “(O^ (18) · [98] In expression (18), = f2(x)~^.
[99] /„„(') has very complicated nonlinear characteristics. Due to impacts from manufacturing defects, external environmental variations, modeling errors, etc., system parameters such as the parameters above may be unknown or less precise, and meanwhile, variations of the system parameters result in chaotic oscillations. In view of the above, it is necessary to search for effective ways to overcome negative factors and nonlinear characteristics in the controller. Therefore, the RBF neural network is used to approximate the nonlinear function fun(-)-qT2 (t) ·χ2 (χι,χ2)+β2 (χ15χ2), wheree2(x[,x2)>0 .
[100] To reduce the computation burden, a number of weight vectors for the RBF neural network is decreased by taking (19;
2018100710 25 May 2018 actions. Using the Young's inequality, the expression is obtained as :
q[ (t)x2 (x,, x2) = — y 2 (t)xT2 (x,, x2 )x2 (¾, x2 ) + ng
2« [101] In expression (19), y2 (/) = |pf (/)¾ (/)||, n2 > 0 , /»(/) =y2 (t)-y2 (/), and y2(0 represents an estimation value of y/) .
[102] For the problem that the derivative of a2 will increase the design complexity and the computation burden, an estimation value of the derivative of a2 is obtained using expression (20) of the extended state tracking differentiator:
(20) [103] There is a nonlinear function:
e/ jl-o.
\hej\^de fal(de,heJ} = '4 ‘ , where b/!,j = l, L,nr and / = 1,2
KPsign(Ae/),|Ae/|><
represent feedback gains, hej represents a tracking differentiator error, de>0 and ae >0 .
[104] A Lyapunov function is selected as (21:
[105] In expression (21), g2>0 .
[106] An extended state tracking differential item v 22 is used to estimate and a time derivative of V2 is given as:
!&=!&+ s2 ( f2 (·) + u(/) - v 22) + — (i)y&2 (/) (23)
2018100710 25 May 2018 [107] It can be known according to expression (20) that he2=v2i-a2. There exists the inequality \v 22--c&,\<ly .
[108] Expressions (18) and (20) are substituted into expression (17) to obtain:
/&<-zrz -cpj -wd -TyJ/jxj (y,x2}x2(y,x2)s2 + d?2 +e2 +u(t)-v 22 +2n
N/ + 4£||P||H2 [109] In expression (23), |e2(x,,x2)|<e2 .
[110] The actual control input (i.e., the adaptive neural network controller) u(t) and its adaptive rule (/) are respectively obtained as:
,(/) = - Ih -7- -2 (07 0 ’ h )h 0 - U)h + v 22 -c22signs2 (24) and
Λ (0 = ^7 (x1,x2)x2(x1,x2)-?2 2 -m2y2(t) (25)
2n2 [111] In expression (24) or (25), c21>0, c22>0, and m2>0 .
[112] Expressions (24) and (25) are substituted into expression (23) to obtain:
I&<-cf -c,f + e,S, \s2\-X/X)y\(t) + ^ + l 82 p2|/ + 4£||7>||2 (26)
In expression (26), y2(t)y/2
Figure AU2018100710A4_D0001
It can be known according to Lemma 1 and the Young's inequality that expression (26) can be simplified as:
:27)
2018100710 25 May 2018 + (-61+0.5)^+(-621+0.5)^--0-^)4(/)1 +74 + -^-^2(01 +^+47 2§2 ^§2 [113] Theorem 1: For control problems of the arch-shaped microelectromechanical system with unknown system parameters, chaotic oscillations, immeasurable states and partial state constraints under the distributed electrostatic actuation, if the extended state observer is constructed as expression (10), and the adaptive neural network controller integrated with expression (22) of the adaptive rule and expression (20) of the extended state tracking differentiator is designed as expression (24), all signals of a closed-loop system are uniformly ultimately bounded, and meanwhile, the output defined as constraint is not violated.
Proof: The Lyapunov function is
Γ = — In ’’ + —+-y 2 (t)+z TPz
Ί k2-s( 2 2 2g/2K/ [114] Then, its derivative may be obtained as :
A=f&<-zrz-(g-0.5)5(-(e21-0.5).f2 22-^(/)1 + do ^~JoV+do [28'
In expression (28), = min (2x(q-0.5),2χ(η21-0.5),τη2) and , fl-, iTl -, ι , , i2 -)
4,=-+-—^2(O| +^L ^§2 [115] Both sides of expression (28) are integrated to obtain:
< V (/) < + t(/0) - 4 »< 4+K(Zo) \ 70 ) Jq (29;
[116] Therefore, and >4(z) are affiliated with a compact set:
wr / /, S2 ,/» (/)) | V < V (t0)+-j-, vz > t(
130'
2018100710 25 May 2018 [117] From expression (29), the following expression is obtained:
>· 2 2dn lims < —
/.,» 1 j •J n :3i) [118] Since y(t)=sl (z)+p , |.v, (f)| < and |xr|<x„, it can be inferred that |y(z)| <T, +x„ =Ti · Therefore, a conclusion can be drawn that the output y remains within the given constraint range. Up to now, the proof of Theorem 1 is finished.
[119] Simulation result analysis [120] The reference trajectory is selected as x. =0.4sin(2z) and the output constraint is defined to be |y| < kcl = 0.43 . It can be seen from the above description that |g (z)| < /y = 0.03 is tenable. The parameters of the proposed controller are selected as c, = 8, c21=8, c22 = 0.1, g2=l, /72.,=10, and «2 = 5 . The parameters of the extended state observer are designed as K, = 1 and /=1. The parameters of the extended state tracking differentiator are selected as /=3 , /=300, de=0.022 , and ae =0.3. An initial value of y2(z) is 0.02 . Moreover, the RBF neural network includes 9 nodes , the weight factor mt e[-6,6] and the width s =3 of the Gaussian function .
[121] Figs. 6(a)-6(c) show the first error function under different R values. It is obvious that the tracking performance between the reference trajectory and the actual trajectory is achieved. Figs. 6(d)-6(f) show that all the tracking errors are less than ±0.03. |y|</ is guaranteed after the symmetric barrier Lyapunov function is used. In addition,
2018100710 25 May 2018 as compared with the phase diagrams in Fig. 2 and the time histories in Fig. 3, it can be known that the arch-shaped microelectromechanical system immediately attains a steady state and its chaotic oscillations subjected to the distributed electrostatic actuation are also completely suppressed.
[122] Figs. 7-8 reveal observer performances of the state observer. It can be seen from Figs. 7-8 that the state observer can precisely estimates actual signals, and reduce the restrictions on physical sensors. Although the arch-shaped microelectromechanical system has the unknown parameters, the partial state-constraint and the chaotic oscillations, observer errors quickly converge to zero with less oscillation.
[123] Fig. 9 illustrates control inputs of the arch-shaped microelectromechanical system under the actuation of static DC voltages and harmonic AC voltages for different R values. It can be seen from Fig. 9 that 4 curves of the control inputs of the arch-shaped microelectromechanical system can remain consistent at a fast rate. Meanwhile, it is also illustrated that the proposed controller has a better anti-disturbance ability.
2018100710 25 May 2018

Claims (9)

  1. WHAT IS CLAIMED IS:
    1 . An adaptive neural network control method for an arch-shaped microelectromechanical system, comprising:
    (a) constructing a system model of an arch-shaped microelectromechanical system based on an Euler-Bernoulli beam to obtain $4) + mfff) + (]. + 2h2a x)q(t) — ( 0 )) (tj + afr (t) + u(t) = 0 m, fr a\' f r anc} bn representing dimensionless parameters, representing a state variable, w° representing a frequency, and representing a control input in expression (1); and (b) constructing an adaptive neural network controller for suppressing chaotic oscillations of the arch-shaped microelectromechanical system and ensuring state constraints of the system, comprising:
    ensuring an unviolated output constraint of the arch-shaped microelectromechanical system using a symmetric barrier Lyapunov function;
    adopting an RBF neural network having an adaptive rule 20 to estimate an unknown nonlinear function with an arbitrarily small error;
    introducing an extended state tracking differentiator to address a need of a virtual control item for repeated derivative calculations in a backstepping control;
    25 designing a state observer to obtain immeasurable state information; and
    2018100710 25 May 2018 fusing the extended state tracking differentiator and the state observer in a backstepping framework.
  2. 2. The method according to claim 1, wherein if ancj are defined as new variables, expression (1) of the system model in (a) is rewritten as:
    ^-=x2,y = xi η ί > \ 6 (l + 2Acos(w0/)) = -mx2 —11 + 2h a{ Ixj H— - 7 + 2a{hx^ —axxx +u(t} (2) , wherein a system output y satisfies a constraint condition , where Ti>0
  3. 3. The method according to claim 2, wherein in (b), the constructing an adaptive neural network controller for suppressing chaotic oscillations of the arch-shaped microelectromechanical system and ensuring state constraints of the system comprises:
    (bl) constructing the state observer expressed as:
    |v= V+ Kfy- 1,) | % = G(i/)+ /, (1)+ K2 (y - 1,) (3) , wherein x t*1’*2! represents an estimation value of [x„x,J ,, V, 4- ί 4- ί Κ=[Κ,Λ,ί L -, and through an appropriate selection of L J „ iK, i«
    5 = £ 1 « e k 0u e - u wherein expression (4) belongs to a Hurwitz matrix;
    (4) ,
    2018100710 25 May 2018 if an error of the state observer and a nonlinear item are defined as z = x~x and . . Z>(l+ 22?cos(Mq)) ,
    f.fx) “ (1+ 2/Γα,)χ,+ -, __ - mx2 - a]X] + 3a,hx(
    Bz + f (x) where
    2b
    7c h- *7 (5) , /(*) = $ AU)- AU)!
    Β , and
    - [zi a! .
    wherein a Lyapunov function is selected as:
    V„ = z Pz (6) , where p represents a positive definite matrix, and satisfies the relationship btp+pb=-i- ancj wherein using a Young's inequality a derivative of is obtained as:
    (7) ; and (b2) constructing the adaptive neural network controller, wherein, wherein an estimation value frhf(X} of a nonlinear item frhf (if) in the system model is expressed by the following RBF neural network expression:
    Lf(x,q(t)) = qT (t)x(X) <7 = Γ<7ι ,<72>l ,¾ Ί . ,, , / ,i where L 'j represents a weight vector, represents a node number of a neuron, XrzR represents an -P 1- x(X)=[xl(X),x2(X)X ,x,(XW eR1 . _ x, (if) , .
    input vector, and v ’ 1V 7 2K 7 p with ‘' > being a Gaussian function expressed as:
    2018100710 25 May 2018
    m.
    χ, (%) = exp [X —mf (X —m)
    2s:
    i = 1,L , I where s·' represents a width of the Gaussian function, and
    i. = ,mln]T represents a weight factor;
    wherein a first error function is defined as si~xi~xrr x,· representing a reference trajectory, and a symmetric barrier Lyapunov function is selected as:
    V, = —In
    1. kJ
    2 kJ — sj 0 (15) , _jy Ls < Zr where b} q , and the constraint condition Nl h' is not violated;
    wherein a value of the variable is estimated using the state observer, due to an immeasurable state variable %2, ;
    wherein a second error function is defined as A’2 , where represents a virtual control;
    wherein a derivative of is given as
    5i (y+a2-j<fe+z2) + (16) , l· -L· - - v2 , Λ1 h,/·, *-’l where 1 wherein the virtual control is obtained as: fl2
    C G> 0 where 1 , to obtain:
    i f <-z ‘ z -c.sJ +^-X+l ‘ ‘ ‘ k, + 4L (17) ;
    2018100710 25 May 2018 wherein a derivative of ‘2 is calculated as where
    L (·) = Λ(χ)-4.
    (18) , the RBF neural network is used to approximate a nonlinear function: -^0) = ¾ (i)x2 (x„x2)+e2 (x„x2), where e2(x,,x2)>0.
    wherein a number of weight vectors for the RBF neural network are decreased by taking actions, and the Young's inequality is used to obtain:
    q( (i)x2 (x,, x2) = — y 2 (t)xT2 (x,, x2 )x2 (x,, x2) + (19) , l20) = |K0k0)||, «2>θ; liO) =12 0)^0), and ^0) i20).
    where represents an estimation value of wherein an estimation value of the derivative of °2 is obtained using expression (20) of the extended state tracking differentiator:
    v&t=v i’&2 =-^,2./^ O^-A) (20) , and wherein a nonlinear function exists:
    fal(de,hei) =
    KP signOq), > d, represent feedback gains, differentiator error, 4>' where ’ n , and '-I2 h
    ej represents a tracking and Ωρ>θ;
    wherein a Lyapunov function is selected as:
    2018100710 25 May 2018 (21) , where “2 >0 ;
    wherein an extended state tracking differential item v 22 is used to estimate and a time derivative of A is given as :
    j&+ s2 (f2 (.)+u (t) - v ,2) + — /«2 (ψ* (f) (22) , wherein v 21 a2 ys obtained from expression (20), and there exists the inequality 22 wherein expressions (18) and (20) are substituted into expression (17) to obtain:
    (&<-zrz-cf +f^^G-y2(t)x[ (xt,x2)x2(xt,x2)s2 + A?2 +e2 + u(t)-v 22 +
  4. 4~2|/+4£||p||H2 where |e2 (x,,i2 )| < e2 _ (23) , wherein an actual control input and an adaptive rule y H' of the actual control input obtained as:
    '0) are respectively ί(ή = -[ c21 +| |ί2 “P’-yriz (ήχ2 (xif2)x2 (χιΆ)Α +v 22-c22signs2 f2n2 (2 4), and ln2 (25) , wherein c2i>()f c22>0^ an(g m2 >0 expressyon (24) or (25) .
    2018100710 25 May 2018
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    2018100710 25 May 2018
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    2/9
    2018100710 25 May 2018
    Fig. 2
    3/9
    2018100710 25 May 2018
    CZ) is
    Fig. 3
    X
    4/9
    2018100710 25 May 2018 jusuodxs AounduXq xssSjpq
    Fig. 4
  5. 5/9
    2018100710 25 May 2018
    Fig. 5
  6. 6/9
    2018100710 25 May 2018
    Fig. 6
  7. 7/9
    2018100710 25 May 2018
    JOAJOSqO JO J0JJ9 lx ‘(lx)jpq
    Fig. 7
  8. 8/9
    2018100710 25 May 2018
    Fig. 8
  9. 9/9
    2018100710 25 May 2018
    LO
    O
    O
    II
    C\ Ό Ό CM
    O
    II cc cc cc cc co
    Ϊ3
    LO
    O
    CM
    O
    CM
    O ’st
    O
    CD
    I
    O
    I
    Fig. 9
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