Anti-oscillation adaptive control method for fractional order arched mems resonator

专利摘要:
The present invention relates to an anti—oscillation adaptive control method for a fractional order arched MEMS resonator, and belongs to the field of anti—oscillation control. In the design process, the Chebyshev neural network with an updating single weight is used to compensate the uncertainties. The Nussbaum function is used to solve the problem of unknown control direction caused by the actuation characteris tics in Caputo fractional calculus. At the same time, a tracking differentiator based on the hyperbolic sine function is designed to solve the repeated differentiation of virtual control because of complicated fractional—order calculation. Then, under the framework of adaptive backstepping control, an anti—oscillation—based adaptive control scheme fused with Nussbaum function, neural network and tracking differentiator is developed in the framework of adaptive backstepping by means of continuous frequency distributed model. Based on the fractional order Lyapunov stability criteria, the asymptotic stability of the closed—loop system is proved. Finally, numerical simulation verifies the validity of the proposed solution.
公开号:NL2024372A
申请号:NL2024372
申请日:2019-12-04
公开日:2020-08-13
发明作者:Luo Shaohua；Liu Zhaoqin；Qu Yongjie；Ge Zhihong
申请人:Chongqing Aerospace Polytechnic；
IPC主号:

专利说明:

ANTI-OSCILLATION ADAPTIVE CONTROL METHOD FOR FRACTIONAL ORDER
ARCHED MEMS RESONATOR Technical Field The present invention belongs to the field of anti-oscilla- tion control, and relates to an anti-oscillation adaptive control method for a fractional order arched MEMS resonator. Background In recent years, as an extension of integer-order calculus, fractional calculus has attracted extensive attention of aca- demic community in the fields of electronic engineering, robotics, bioengineering, and signal processing. It has poten- tial advantages of robustness, design freedom and transient performance, can accurately describe actual engineering ob- jects and processes and is widely used in control systems. A micro electro mechanical system (MEMS) resonator has been widely used and has received wide attention due to their sen- sors, microvalves, switches and filters. The MEMS resonator has high nonlinear characteristics such as parallel plate forcing, midplane and squeeze film damping. These characteris- tics may cause chaotic oscillation, which is undesirable and may cause adverse reactions. At the same time, due to changes in the external environment and the existence of manufacturing defects, new challenges such as fluctuations in feature param- eters, mechanical coupling, and dangerous noise will be faced. With the popularity of the fractional calculus in engineering and the high quality requirements of system performance, the openness problem of anti-oscillation adaptive control of the fractional order arched MEMS resonator needs to be solved ur- gently. In order to stabilize the unstable periodic orbit of the chaotic system, Ott, Grebogi and Yorke have proposed the OGYmethod at first.
Then, for the synchronization and chaos con- trol problems of the integer-order system, research scholars have proposed many effective methods to achieve system stabil- ity, such as adaptive control, backstepping control, sliding mode control, He control and contraction theory.
The drawback is that these methods are limited to the integer-order system without actuation characteristic, and whether they can be di- rectly applied to the arched MEMS resonator remains to be further studied.
The chaotic system is more accurately modeled by fractional calculus theory.
For a long time, fractional or- der chaotic systems such as Chua's circuit, Rssler system, LU system and Van der pol-Duffing system have been reported in the field of fractional calculus, but do not involve incentive and constraint.
In practical engineering, the actuation char- acteristics including input deadzone and saturation are unavoidable.
Ignoring such characteristics may cause system instability or performance degradation.
Some researchers have discussed a 2D torsional MEMS micro- mirror second-order sliding mode control solution with sidewall electrodes.
The controller is composed of equivalent control and switching control to solve the uncertainty and ex- ternal interference of the model.
Salaff et al. have applied bandpass SMC technology to the drive mode of the MEMS resona- tor and resonant frequency sensors.
However, inherent jitter related to SMC cannot be completely suppressed.
At the same time, these researchers are not dedicated to the problem of fractional calculus and actuation characteristics in the MEMS system.
Backstepping control is one of the effective tools for the design of fractional order nonlinear system controllers.
In a fractional order inversion framework, Ding et al. have solved the pseudo-state stabilization problem for a symmetric fractional order nonlinear system with unknown parameters and additional disturbance.
Das and Yadav have researched thechaos control and function projection synchronization problems of fractional order T systems and Lorenz chaotic systems by the backstepping method. Wei et al. have proposed an output feedback control method based on adaptive backstepping for nonlinear fractional order systems. These methods rely heavily on accurate system modeling and cannot handle unknown nonlin- ear functions in dynamic models. As the system divisor increases, "system explosion" phenomenon related to the back- stepping will occur.
Summary In view of this, the purpose of the present invention is to provide an anti-oscillation adaptive control method for a fractional order arched MEMS resonator.
To achieve the above purpose, the present invention pro- vides the following technical solution: An anti-oscillation adaptive control method for a frac- tional order arched MEMS resonator is provided. The method comprises the following steps: Sl: establishing a dynamic model of a fractional order arched MEMS resonator with unknown actuation characteristic by using Galerkin decomposition method; and SZ: designing an adaptive controller.
Further, the step S1 is specifically as follows: the dynamic model of the fractional order arched MEMS reso- nator with unknown actuation characteristic is written as follows by using the Galerkin decomposition method: | cis tja £3 Ee Be ost] valies be sss {13 hile hex te weg Pe vie {ben ie} Ate], where variables are defined as follows:
‚Pe ; t= dd, fm, ; = represents a ratio; NY X11 SII represents a dimen- “pe : : : a= Cain fh, : sionless time variable; *” vYS1/Vi represents a damping ein: y=8_ si jb, : coefficient; 1 m~ily Vil represents a stretching parameter; Wo EEE jj EE ile } Jun dT Sie = ln AE *“*» represents a voltage parameter; oe = "ay ‘iu Vii repre- gts A. w POE Ca , Bel gd sents frequency; #4=h/g, represents an initial rise; LE represents a constant; x=q(f) represents displacement; =x, =. 451) represents speed; a represents a fractional order; ( repre- sents a symbol defined by Caputo in the field of fractional calculus; (4) represents an unknown actuation characteristic; AX on r= represents a length coordinate; ¥ represents a first nor- Ww, Teng malized mode shape; Ww,=— represents deflection; Te &£y | | DE ( represents dimensionless quantity; FE represents time; Et ere ease DS EN ee MES Sel NE { SERIE U ij Ue JE pap WET) = —g{thel{s} HT gE AEE LEY represent . ~ bh od constants; . represents actual control input; b=—, d=--, &o £y ‚Gt L 4 ‚ We k,>07 represents length; < represents cross-sectional area; b represents width; € represents a viscous damping co- Co = efficient; ¢ represents thickness; © represents Young’s 7 . . modulus; +” represents moment of inertia; p represents mass density; £, represents harmonic load frequency; %s¢ representspermittivity of vacuum; Fae represents DC voltage; Fa repre- sents AC voltage; "¢ represents arched displacement; wy represents actuation frequency; an asymmetric non-smooth saturated nonlinear actuation 5 characteristic M(u) exists in the input and is expressed as 7 HH, | Lua, {1}, a {izes Hy (a) = 0, vim {tina {2 : {a +8 {)) CB SS a | 9 Kivi, , (2) where fj and 7 represent limits; a(t) and a,(f) represent time- varying functions; J and /, represent functions of deadzone characteristic; Zim; and 4, represent positive unknown break- points; because a{(f) and a, (f) are time varying, asymmetric, satu- rated and non-smooth, a smooth function is introduced to approximate asymmetric non-smooth saturated feature M(u)=S(u)+D(u) (3) and ye se ope THN Su) = mom (4) where w represents a design parameter; D(u) represents an approximation error and iDia)| = Mu} Sla) SF; I represents a positive and unknown constant; according to mean value theorem, for the smooth function Su) : $= 500) DD (5)
by defining p=aS(u)/ou and obtaining S(0)=0, (3) is rewritten as: M(u}= pu+D(u) (6) a (ry and a,{(f) represent time-varying functions and can re-
3 flect the actual conditions of a nonlinear system under internal and external interference; such smooth functions only need the upper and lower bounds of the actuation characteris- tic; different values as approximation coefficient w result in different approximation results for M (uy
system parameters are selected as ¥ = #893, p=03, u=0.1, = 088, §=1199883 and ww, = 34706; the chaotic oscillation of the fractional order arched MEMS resonator is revealed through different fractional orders and driving amplitudes with the help of a variable step size START/TR BDF2 solver; transient chaos occurs at fractional order values such as a=l10 and 095; then, the arched MEMS resonator suddenly switches to a non- chaotic state at «=09 and 0.75; definition 1: the Caputo definition of f(f)in the fractional derivative is expressed as | . fi 5} § 5 } | / i : en of _ SE SY OA FEE) Cop J Uh ee LE } (7) where Fim —a) =| et" =Tldt represents a gamma function; , and f(t) represent an integer and an „-order derivative of f(r) lemma 1: for continuous functions f(r) and f(t), the follow- ing equation holds
HEL AES AU RD 8 (0) OY { ay I df, {1 } Lr & df; {s } ir} FR CTI YT LL, TTI 4 where O<ax<l; the lemma 1 and a relational expression =r Ee zo ienie ie are used to obtain: 5 == can (0-2 (0) if (1) <0 (9) where J()=f(f)=f(D; lemma 2: fractional order system „S&/(t)=O(tf) and O<a<l; HAER En . . . HO eR and Gif ER are converted into a linear continuous fre- quency distribution model of a fractional order integrator; Su {aneh Ny ew fo = = oF, £] + Ot), Ue) = Ja le) ¥ eo, dn, } (10) Cy sn iaad ‚ . Ui ‚ ‚ ‚ Blea fr ER where Bale) „*x represents a weighting function; Flat} ER represents an actual state of the system; definition 2: if the function N(h) meets the following propertiestim sup {* (1) dh = +o0 11 Hm sup i Í N{fydh = { ) ee lim inf 7 {, N (&)dh = —oo (12) the function is called as Nussbaum function; Nussbaum function is regarded as an effective tool for dealing with an unknown symbolic problem in a actuation char-
acteristic; the following lemma related to the Nussbaum function is introduced to facilitate controller design and stability analysis;
lemma 3: it is assumed that V()and #()are within [0x), there is a smooth function of ¥V(t)20, and N()is a Nussbaum function; then the following inequality holds | | (13)
where C,>0; g(t) is a non-zero constant; J represents a proper constant; then ¥V(f, nt) and Jp #(0) N(s)e Dime are bounded;
assumption 1: a reference trajectory x, and a-order deriva- tive are known and bounded; meanwhile, state variables x{(f) and x{(f) can be measured; for the fractional order arched MEMS resonator with uncer- tainty and time-varying actuation characteristic, an adaptive control solution is proposed, so that output y=x(/) follows the reference trajectory x, at a slight error; and at the same time, oscillation related to chaos behavior and asymmetric deadzone is completely suppressed.
Further, the step S2 is specifically as follows: Chebyshev polynomial is selected in the form of two recur- rence formulas u Ta (X)=2XT(X)-T_ (X). T,(X)=1 (14) where yep and 7,(X) are defined as x, 2Y, 2X-1 OF 2X41; for [et] ER”, a strengthened form of the Chebyshev polynomial is constructed as: ò (xX = 1 s Ty (x1) $0 Ta {xg} s 7 ‚it Gt) 7 ‚Ds {xw) i (15)
where T;(xy),1=1,..,n,J=1,..,m represents Chebyshev polyno- mial; §&(X) represents a basis function vector of the Chebyshev polynomial; “represents the order;
for any given unknown continuous function f(X) on a compact set, F(x) is fully accurately approximated to the continuous function based on a general approximation theory of Chebyshev neural network to obtain FlOy=o7 (nex) (16) where f(f)is a smooth weight vector; there is a Chebyshev neural network Fade : (17) where SX) zg is an approximation error; fg and p, respec tively represent proper boundary compact sets of ¢(t) and x; an optimal parameter ¢° is set to be equal to arg min ze IEX) — Fix.) | die) = pt) gp" (5) Where p' is called PEN [Yesy as artificial quantity; when z= 9 then ls{fjjas: to promote quick on-line computation, the following conver- to sion is used to reduce the number of weight vectors of the Chebyshev neural network GLA) Fh CENA er lon 16 where Ae) the relational expression Alf} = A406 At} holds; Aft} is an estimated value of A(t); 5 is a small positive constant;
a mathematical transformation of the Chebyshev neural net- work related to the number of weight vectors is derived by means of Young’s inequality;
step 1: defining a first intermediate variable (19)
where the tracking error eff) is defined as e(f)=x{)-x,(1); #4 represents a positive design parameter; if Z‚()—>0 and vio, 1 then e(£)>0, [anya >o, Vi; Lal a second intermediate variable e,(r)=x,(f)-a,(t) with the ‚ Tey — a £ Yo a FE Hd oh ‚ 7 tracking error Z(t) zelf} + a | elijdr is selected, where oy > is a design parameter and a,(t) represents virtual control; a derivative of Z‚(f) is derived from the definition of Caputo fractional calculus; cin Zlij= Zl) rat} a}. elders xl) (20)
virtual control is selected as: a {em =A Z tijen, {oe {dv op ijs Sx) 7 wa ( 2 1 ) where k >0represents control gain; based on the lemma 1, the following continuous frequency distribution model is obtained: Vor + x ; SF Wig £1 S © N ~ y N 3 k vi x y 3 STEIN $ nll oe dio eS LE a {Ey Iz, {2dr eon bo S050 LL 3 De: HE go ; yf yy Lyapunov stability criteria are considered ES £3 ana ! w fo) je {. 3 3 Foor § {3} rd En {ied BY {dd FE (23) sina} i {ur} IE veele . where *a ws ae 7 a time derivative of V{f)is calculated SESE FY oh EE fee 2 PIE LT FY Ee) gr Fy {¢} Ss ne WHE {a } YW, Leds ws Kf {¢} Ee {4} & {2} (24) step 2: selecting fractional order Lyapunov stability criteriaoss elder [a (9) ft) ds #1) ’ (25) where h,>0, th, (10) = 22 and teln) =>; Z(t) is differentiated to obtain FES pe iele ls ft) (26) HE {e200 (0) ls) eileg le) re} where | aylik] ;
f() is a high-order nonlinear function, where system pa- rameters such as #4, a, y and &, cannot be accurately measured, and it is very difficult to establish an accurate system model due to the influence of inside and outside factors; differentexternal actuations generate harmful oscillation for the arched MEMS resonator, and such oscillation reduces the per- formance of the system to a certain degree; to solve these problems, the Chebyshev neural network is used Silesis ie sa dae is Elisa Ula radi) (27) actually, .%%a,(f) cannot be directly solved due to calcula- tion complexity; to solve this problem, a fractional-order tracking differentiator based on a hyperbolic sine function is designed to estimate the fractional order derivative of the virtual control a,{(f}; RE = Tay = Giz, =r a sinh (4, (=, 1-4, (1) +¢,sinh(d,z,, )/ a Hef, 2 2 2, (28) where the states +, and .%jae,(t) of the tracking differenti- ator based on the hyperbolic sine function are equal; r>0, ¢>0,i=12, and d >0i=12 are design constants; a relational ex- pression fim [|2,, —a, (Ne =0 with positive 7 exists; (27) and (28) are substituted into (26) to obtainiran ee FL rs w Ce ow ded EN dl} ) wR gy {¢] a ie bs ) Wily Sys PEE if } (29) Ce RATE TO IVS can be easily derived; a continuous frequency distribution model is further derived by using the lemma 1; sss (osje pere QI eos Peat Te li} a 2% Sp FEE FREE IT Ke] | sr SA $e SN ave & 2 Z. {fe | Ha, Le } WAtard } gi
NT RTT ~e Lend Pp SAE (30) ey RE xy 3 Ng Atje) a VW, Led ida © Ni i Eo ” 5 the time derivative of (25) is taken . | Cw A {OX (ey {2,2 Ke Vee} Cow fo Yop Reg Cond fo fehl ee Flos = § wn EN a8; iet 3 pr 5 3 TR at i A Joa #3 er) fen ¥ Les of if 3 ye à 3 «3 - J, os Ltd ee SZ cd) Gls lee) lee) Eide ; TARR 2 VS As &, i it, 38 3 3 3 12 * © 3 Gl) fe bey where ; the Nussbaum function is used to construct the following control input Oe Zaki elite) . Ce RN pee & ard wh Bf Adis es Ja en] ik sien A le} | where k,>0 and k,>0 are control gains, and has update law SA led Le {ene {ee Wiljeg iz 1 Ce AA 2 he Go $86 JN ISLE FO TEI RA LE (33) Cd ZAR aE (ee ed : mn Yn | & EH = A, U i “Roy {£} = ZARA en ln) FL { £3 wk, ied, is} Pis A, {¢ i. i ke HUH Rafe TR HATE BERS US So dig (34) where g, is a positive number;
the update law and the control law are substituted into (31) to obtain V,(t) Co Eee | {a {we} SI [stie A Flas -2 Vi Lah {oa Jur eeen vo Sr {aN {ng} 1} de . 3 ( 3 5) A os > oad LN REN & VEA JA, if } > ki PEE Sls ië } * DN theorem 1: a fractional order arched MEMS resonator with 3 unknown actuation characteristic is considered if the assump- tion 1 holds; if the anti-oscillation adaptive control method (32) composed of the adaptive rates (33) and (34) is intro- duced, then all internal signals are kept bounded and oscillation including the chaos behavior and the asymmetric deadzone is completely eliminated; proving: the entire Lyapunov candidate function is defined ¥ uw 3 > w i {; a} 3E 3 {: 3 i 3 ot un | * if [a } ‘jj ¢ { Lf iki Co (3 6) = Leds Hs LM Ni] Ee 3% Is Mg, Re da Wd Rau di Eran oF Sne ob ome he following i sccording TO gd (Al) ziee + Lal TRC Tollowing 1s obtained Co Ioan Co { Of, {ie} Tl foot ke oo EN Fs we {ed ¥ fede ee Sl ONT} VAST] BARTER i gr HADNT ER oe A. vo RS HARZ G7) where amen BES ee + by defining e= mink, 2k, 6.) the above formula is simpli- fied into Mods) ain (aN bi) (38) both sides of the above formula are multiplied by eSit to obtain
4 { i in) EN à | : == u . 9 rosin De Hom «4 BN { B] } sh 1 RE SL ar (3 ) uit OR PV ARES 3 by defining «=S the above formula is integrated Ax o Z(t), Z(t) and &,{¥) belong to compact sets 5 . {i vp of x ee of Say sE zn Tae Sd ee Al Qs OZ OA OW Me] apron Ghi)eds) (41) therefore, all the signals in a closed-loop system are bounded; the following is further proved gs if a ay 4 2 Ee U = 2g, (42) Et at this point, proving of the theorem 1 is completed.
The present invention has the following beneficial effects: 1) An updated single-weight Chebyshev neural network is proposed for the uncertain dynamics and disturbance problem in the field of Caputo fractional calculus. It can facilitate quick on-line computation and reduce the requirements for dy- namic control equations. The Nussbaum function is used to solve the uncertainty problem of the control direction caused by the actuation characteristics, and the vibration of the control input within the threshold of the asymmetric deadzone is better avoided.
2) A tracking differentiator based on hyperbolic sine func- tion is designed to overcome the "system explosion" problem in traditional backstepping technology. The virtual control dif- ferential item has higher estimation accuracy than the first- order filter and the general tracking differentiator.
3) By using the principle of adaptive backstepping method based on the continuous freguency distribution model, an anti-
oscillation adaptive control solution that combines the tracking differentiator, the neural network and the Nussbaum function is proposed to force the system state to approximate the reference signal with very small error.
At the same time, for the frac- tional order arched MEMS resonator, the anti-vibration purpose near the resonant frequency is achieved in the actual control within the asymmetric deadzone threshold, which reduces the con- troller vibration.
Description of Drawings To enable the purpose, the technical solution and the bene- ficial effects of the present invention to be more clear, the present invention provides the following drawings for explana- tion: Fig. 1 is a control schematic diagram of an arched MEMS resonator; Fig. 2 is a actuation characteristic under asymmetric non- smooth saturated nonlinear influence; Fig. 3 shows phase diagrams under different fractional or- ders; (a) is 1.0; (b) is 0.95; (c} is 0.9; (d) is 0.75; Fig. 4 shows phase diagrams under different actuation am- plitudes; (a) is 0.01; (b) is 0.02; (c) is 0.1; (d) is 0.21; Fig. 5 shows position tracking; Fig. 6 shows velocity tracking; Fig. 7 shows control input before and after actuation char- acteristic Fig. 8 shows Nussbaum function under different actuation amplitudes; Fig. 9 shows intermediate variables under different actua- tion amplitudes; (a) shows an intermediate variable Z(r).i=1 under different actuation amplitudes; (b) shows an intermedi- ate variable Z(f).i=2 under different actuation amplitudes;
Fig. 10 shows control input under different actuation am- plitudes and fractional orders; (a) shows control input under different actuation amplitudes; (b) shows control input under different fractional orders; Fig. 11 shows updating single weights under different actu- ation amplitudes and fractional orders; (a) shows updating single weight under different actuation amplitudes; (b) shows updating single weight under different fractional orders; Fig. 12 is a phase diagram under different actuation ampli- tudes and fractional orders; (a) is a phase diagram under different actuation amplitudes; (b) is a phase diagram under different fractional orders; Fig. 13 shows intermediate variables under different frac- tional order values; Fig. 14 is a tracking state Z,; under different actuation amplitudes and fractional orders; (a) shows anti-oscillation performance under different actuation amplitudes; (b) shows anti-oscillation performance under different fractional or- ders; Fig. 15 shows control input under a different solution; Fig. 16 shows a state Z;, of a tracking differentiator un- der a different solution; and Fig. 17 shows a second intermediate variable under a dif- ferent solution.
Detailed Description Preferred embodiments of the present invention will be de- scribed below in detail in combination with drawings.
1. Problem posing and mathematical foundation The arched MEMS resonator is composed of DC, AC, a bottom electrode, a double-clamp arched sharpened micro beam, two an- chors and an integrated operational amplifier. Apparently, thebasic resonant frequency of such arched MEMS rescnator is higher than that of a fixed beam or a cantilever beam.
Fig. 1 shows a control schematic diagram of an electrostatically driven arched MEMS resonator.
When the gap distance is g, and the bottom electrode placed parallel to the X-axis is oper- ated, electrostatic drive is performed.
The quantity of physical dimension is expressed as |; the dynamic model of the fractional order arched MEMS resonator with unknown actuation characteristic is written as follows by using the Galerkin de- composition method: edna dele ad : Cen Asi costo {edn id so TEE I sss {13 | Sid hex deh Lea leben ee MeL where variables are defined as follows: : Fe 3 F = Jk Ja : =p represents a ratio; © = Ti Wy; represents a dimen- pC í í í ev — a i “ en í sionless time variable; # = {Mid represents a damping Co vre =e Ih . coefficient; Y= 585%; jVys represents a stretching parameter; a Yel & . ; . oan = 8 Fane fF . ne gid, represents a voltage parameter; ag = py, fy, repre sf gt . ‚ . ‚ 8, in } { i : sents frequency; h=h/g, represents an initial rise; © id’) represents a constant; x=¢(f) represents displacement; x, =.5q(f) represents speed; a represents a fractional order; C repre- sents a symbol defined by Caputo in the field of fractional calculus; M(u) represents an unknown actuation characteristic; xX . . Feo represents a length coordinate; ¢ represents a first nor- ‚ OW . Ve ar malized mode shape; %,=-* represents deflection; 27, &oay x wy Lien represents dimensionless quantity; Velg represents time; ~ Spt Se IN 3 { ES vg af I . x i" 3 en i a * EN on a ny WEE EE KAT “1 Wl any Et FEES 3 , WE, Ti _ gl t} mie} , Hy : i ES {¥ 3 8X , | 4 dE i Tr ep re- ~ ~ ~ ~ : pn b ° d sent constants; 4 represents actual control input; b=—, d=—, Eo oy dS, so. ; : > Other parameter symbols of the arched MEMS resonator are given in Table 1. Table 1 Representation of System Parameters Param- . Param- . / Symbol (Unit) Symbol (Unit) eter 7 eter ’ Cross-sectional Length (m L gth (m) 4 area (m ) Viscous damping b Width (m) C, coefficient (N:s:m%) . ~ | Young’s modu- d Thickness (m) E ins 8 lus (Pa) 7 Moment of inertia , Mass density > (m*) (Kg'm™) Q Harmonic load fre- Permittivity of quency (Hz) “sd | vacuum (F/m) DC voltage (V) AC voltage (V) ’ Arched displace- Actuation fre- W, wy j ment (m) quency (Hz) In practical engineering, the actuation characteristic is unavoidable.
Its occurrence may cause inaccuracy or system in- stability.
An asymmetric non-smooth saturated nonlinear actuation characteristic M(u} exists in the input and is ex- pressed as
VI oa | { [u wa), a {fis Be M= 0, EE (1) | Liere ti}. on, Sese li) ip Hl, (2) where 7] and 5 represent limits; ¢(1) and «(1) represent time- varying functions; +4 and £, represent functions of deadzone characteristic; a, and 3g. represent positive unknown break- points.
Fig. 2 represents the structure of the actuation charac- teristic. Because afl and a(t) are time varying, asymmetric, saturated and non-smooth, the design of the controller is very difficult. For this problem, a smooth function is introduced to approximate asymmetric non-smooth saturated feature M{(u)=S(u)+D(u) (3) and Ga ge WS Sul = moe (4) where w represents a design parameter; D(u) represents an approximation error and Biu} = Mí(a) —S{u) x FP; FP represents a positive and unknown constant; According to mean value theorem, for the smooth function Su) St) =s(0)+ 50, (5) au by defining p=êS(4}/2u and obtaining S(0})=0, (3) is rewritten as: Mu) = pu+D(u) (6)
Remarks: gi) and g(r) represent time-varying functions and can reflect the actual conditions of a nonlinear system under internal and external interference.
Such smooth functions only need the upper and lower bounds of the actuation characteris- > tic.
Different values as approximation coefficient w may result in different approximation results for M{(u); which is more suitable for the actual system.
Fig. 3 shows phase diagrams under different fractional or- ders; (a) is 1.0; (b) is 0.95; (c) is 0.9; (d) is 0.75; Fig. 4 shows phase diagrams under different actuation amplitudes; (a) is 0.01; (b) is 0.02; (c) is 0.1; (d) is 0.21; and system pa- rameters are selected as ¥ = 7.933, h-03, u=01, a =658, & = 119.9883 and wg = 0.4788, The chaotic oscillation of the frac- tional order arched MEMS resonator is revealed through physical situations such as different fractional orders and driving amplitudes with the help of a variable step size START/TR BDF2 solver.
In all cases, the fractional order arched MEMS resonator presents unstable attractors with cha- otic transients.
In Fig. 3, transient chaos occurs at fractional order values such as @=L0 and 095, Then, the arched MEMS resonator suddenly switches to a non-chaotic state at a=09 and 0795, Fig. 4 shows that the actuation amplitude changes cause different chaotic motions.
Because chaotic os- cillation has the characteristics of randomness and unpredictability, if measures are not taken to overcome the characteristics, the performance of the system will be de- graded.
Definition 1: the Caputo definition of f(f)in the fractional derivative may be expressed asi Praal | Leds, Bias vere Fg i - J { Ho gk © ij Xx A & SEF { I } ws 4 A s (7) ’ © 7 i an i ren FU), JEM Rr Lef Ì where u ‘or eet represents a gamma function; Fin—a)=] ee" 7" gr Pp 9 ron and fn represent an integer and an „-order derivative of f() Lemma 1: for continuous functions f(r) and f(t), the fol- lowing equation holds cds I SEY SY 1 sj vale OS LV ak SNE si Je {1h ijf Ul is {2} 1, {ed A 4] SERT TER UTTER TEN ds i iva} py) is iii} £0 ier} (8) where 0O<a<l; the lemma 1 and a relational expression 1 In we 1° 5 [g m8 ia Cp—peyi—®] =F are used to obtain: (OF (De if (1) <0 (9) where f()=/f"()=f (0; lemma 2: fractional order system „3()=O(f) and 0<a<l; HSE and DIER are converted into a linear continuous fre- quency distribution model of a fractional order integrator; De ce £3 Pe a x _ = iele, £) + Oi), Fx Fo ¢ ET Fa . UE = Pnt), de. (19 oo givin) ‚ ‚ ‚ ef a Tm where g {wm} ==—— represents a weighting function; Fist eR LTT 7 represents an actual state of the system.
Definition 2: if the function N(h) meets the following properties . lp, lim sup“ | N (A) dh =+0 (11) I k& 0lim inf ! [Nan = (12) The function is called as Nussbaum function.
Nussbaum function is regarded as an effective tool for dealing with an unknown symbolic problem in a actuation char- > acteristic. The following lemma related to the Nussbaum function is introduced to facilitate controller design and stability analysis.
lemma 3: it is assumed that V()and #() are within [0x), there is a smooth function of V({t)20, and N(} is a Nussbaum function; then the following inequality holds Voss te] zl) N (ge ndr ve Of mdr J 8 13) where C,>0; g(t) is a non-zero constant; EH represents a proper constant; then V(r), nt) and [st NG DS nde are bounded.
Assumption 1: a reference trajectory x, and n-order deriva- tive are known and bounded. Meanwhile, state variables x(f) and x,(t) can be measured.
The control purpose of the present invention is as follows: for the fractional order arched MEMS resonator with uncer- tainty and time-varying actuation characteristic, an adaptive control solution is proposed, so that output y=x(/) follows the reference trajectory x, at a slight error; and at the same time, oscillation related to chaos behavior and asymmetric deadzone is completely suppressed.
2: Design of adaptive controller The Chebyshev neural network has strong function study and approximation capabilities, and is widely applied to control and modeling of the nonlinear system. The Chebyshev neuralnetwork is used to deal with unknown functions. Chebyshev pol- ynomial is selected in the form of two recurrence formulas L,(X)=2XL(X)-T,(X), T,(X)=1 (14) where yep and T,(X) are defined as x, 2X;, 2X_-1 OF 2X41; for fx A ITE Rr 2 strengthened form of the Chebyshev polynomial is constructed as: EQ =11, Tix), Tao) Tie) Ted T0219) where T;(xy),1=1,..,n,J=1,.,m represents Chebyshev polyno- mial; &(X) represents a basis function vector of the Chebyshev polynomial; "represents the order.
For any given unknown continuous function f(X)on a compact set, fÎ(X)may be fully accurately approximated to the continuous function based on a general approximation theory of Chebyshev neural network to obtain mA Pe > Fld t= ELN) (16) where p(t) is a smooth weight vector; there is a Chebyshev neural network where €{X}>0 is an approximation error; fy and D, respec- tively represent proper boundary compact sets of ¢(t) and X; an optimal parameter fd’ is set to be equal to arg min su Fix FX 2 ; dl) =f) —¢* {(£), where ¢° is called PES [Yens as artificial quantity; when #20, then sl) xs, To promote quick on-line computation, the following con- version is used to reduce the number of weight vectors of the Chebyshev neural network fg Trt IRIN Ts L,8 (18)
where 3 | *; the relational expression Al) = At) holds; At) is an estimated value of A(t); » is a small positive constant.
Remark 2: a mathematical transformation of the Chebyshevneural network related to the number of weight vectors is de- rived by means of Young’s inequality.
Such transformation can increase the on-line solving speed and reduce the difficulty because it only need one weight.
Step 1: defining a first intermediate variable Zia {thro lady 1 0 LN} 3 & 3a vod (19) where the tracking error e(t) is defined as e(t)=x()-x (1); © represents a positive design parameter.
If Z{(1)—0 and Vt>o, then e(1)>0, Île (dt >0, Vio.
A second intermediate variable e‚(f)=x,(f)-a, {tf} with the Fay Fay 2 rt Paid ; tracking error Zit} =e itl +9} elt}ar is selected, where o »0 is a design parameter and a,(!) represents virtual control; a derivative of Z(t) is derived from the definition of Caputo fractional calculus; (20)
virtual control 1s selected as: a {rhe 82 {ahr oe (dy ow dre Sx) or (21) where k >0represents control gain; based on the lemma 1, the following continuous frequency distribution model is obtained:
ded w van, lie ti + Ee { } + EA { + } ag | SN { f dr + TE { $ } en SR Xs { Î 1. 2 {2 vs ee Cy . 1 { tH} = { Za, fw} ¥ Ls {3 den, 22) Lyapunov stability criteria are considered ¥ {4} > “ de a LW} dg {ited FO (23) inf ayn th, {ard minind where ai Ww. i | oi ni), A time derivative of is calculated RE Je 5” Sy 27 SEEN Ler fF Ea i {+ } Ss 1 WHE {ic Blo tdk {¢} + EA } Zr } Lorry DA, dn Sey Sg ik E ITTY OAT ZAT IA (24) Step 2: selecting fractional order Lyapunov stability criteria te dn ¥, ie) ze | i {a } Fe 3 {is ijd aib ZE | Es {i} qd i { add 3 das F Ld } des RE SE * (25) EE . Said where h,>0, gy (0) == and pa lw} =—"; 2 7 En 2 WET
Z(t) is differentiated to obtain ASZ lie pe Sade ef) ft) 1 a: > } # $C Br 4 } SEN } Si { } (2 6) Co nL Sven set, - . ce Oe Ai} HEE jk {1} DON Pea vyd 3 gate 3 rt }s oe} where i Ar . Apparently, Al) is a high-order nonlinear function, where system parameters such as 4, #, y and bh, cannot be accurately measured, and it is very difficult to establish an accurate system model due to the influence of inside and outside fac- tors.
Furthermore, different external actuations generate harmful oscillation for the arched MEMS resonator, and such oscillation reduces the performance of the system to a certain degree.
To solve these problems, the Chebyshev neural network is used
(27) Actually, “4fa,(t) cannot be directly solved due to calcula- tion complexity. To solve this problem, a fractional-order tracking differentiator based on a hyperbolic sine function is designed to estimate the fractional order derivative of the virtual control a,(f); ei 7722 oy =—r [e sinh (d, (za, (#)}) +e, sinh(d,z, ; ) / | 28) where the states z,, and .§a,(f) of the tracking differentia- tor based on the hyperbolic sine function are equal; r>0, c>0i=12, and d >0/=12 are design constants; a relational ex- pression lim | |2,, a. (Nd =0 with positive T exists. Remark 3: 1} Compared with the traditional backstepping algorithm, the tracking differentiator based on the hyperbolic sine func- tion proposed by the present invention can solve the problem of complexity increase. At the same time, the tracking differenti- ator can solve the problem of poor accuracy of a first order filter related to the dynamic scale control and is suitable for systems having any input signal a, (f).
2) The regulation rules of the parameters of the frac- tional order tracking differentiator in practical application are summarized. (a) +; ¢ and d directly determine the conver- gence speed and accuracy. However, too large ¢ and d will cause overshoot. (b) ¢ and d, also affect the convergence speed and accuracy. However, too small ¢ and d, will cause undesirable overshoot.
(27) and (28) are substituted into (26) to obtain We {¢ } w JHE & | {1 } & { Es } BE jg WER {1 } (29)
se dN SEs N $0 eld ijze SUR) l VVB SAS LM As can be easily derived; a continuous frequency distribution model is further derived by using the lemma 1; Piet [a Je a Klee pra ila, Pee}, £3 Py Chae be Fon ld es il in ee FW. ind ie ad, UJ | Nn, VERS IER
A i REC wh ao HE i { Ad : Pe He A { Ei 3 ‘ ( 3 0 } | i dy LIE + i EN go u & ©. PALE a WO {aad we Lí 3 a va * the time derivative of (25) is taken a wr &, iz 3A {ede {sy + 1 ie +5 } Se | Ae sds sa Emy / . . . { fae + EN ES a S$ ally if } i FAs Da wie, le Lod dw + £, HH IH Ti oy { R Loe Fd i La MAAR i i ws, {wile {ass ¥ wo v EY ie} zi {e318 ee ig, ee, 3 os EN 4, Fa &, a {FY &, ThA ù, IE i ‘+ 2 ° > ) ’ = 31) EE a a where . The Nussbaum function is used to construct the following control input GL ZA Eg lee dune) . me es) wa {ahi ei ii) als ess ns] Rr {3 en Lo {4} tia aA ded (32) SH Res U, 3 FAT RAPE TR TAA 32 where k,>0 and k,>0 are control gains, and has update law cdg on LE FRA een ELE BL LEE EE TEE Ws ij ’ LRAT ST 2 sj AA Ds 3 ARETE TS ~ 3 WILT ENTS i 33 i Jk » i ( ) ny Pr ~ 4 2, i 14, ) 2 { & Ay } £, le > En } > x . > y or i . EN pad 2. i H ek { £3 LE AE A I es) LE { £} ok SERA i} FE ë. {2} . ì “HR AH Rada EN FRUIT Ep SERGI fa Tl] (34) where g, is a positive number; the update law and the control law are substituted into (31) to obtain V,(t)
Ay Sy por awe Ns u ids, {aw } ¥ is tii Le ¥ { i} 5 >» i Sj, {il Ww, hie Hr A] ad La) +1} ~ Yen EE eed a SAAS I RE Te = (35) Theorem 1: a fractional order arched MEMS resonator with unknown actuation characteristic is considered if the assump- tion 1 holds; if the anti-oscillation adaptive control method 3 (32) composed of the adaptive rates (33) and (34) is intro- duced, then all internal signals are kept bounded and oscillation including the chaos behavior and the asymmetric deadzone is completely eliminated.
Proving: the entire Lyapunov candidate function is defined EE I lea (36 ond} 3 rk J REY > &. Sa VN Ei EN > according to > Sa Ear t 2352, The following is 9 go DA (EE gods] + ce dale) 9 obtained Lo 2 ze ) .. | | id {bY {a § ies . , í ni sn : #1 * So oe i oem SN x mad No EN 37 gay {dy ie) “Bd U Popi } + > ( ) ci Foard EY 3 ESL wow SL. 23 Se Be ipN{nl eis TE ESET Sek Zier where Lozen , BE €p = — CART By defining eo, = min{k, Tk ge the above formula is sim- plified into > FY eo xt TE s ie Ne f Af eN 3) Loe 38 ¥ {1} x iP {1 } Te Hn - { Fei {1} 3 1 } + ( ) both sides of the above formula are multiplied by e:t to obtain af vies St . 2 0 # U {4 & } ~~ a, Foard NLA et (3 9) somm— Rig ER 3 1 FA in) + i je Fg Gui of è .. Ass 3 Lf # A by defining : _ 5, the above formula is integrated x Szot (40) Z(t, Z,(1) and +:{#) belong to compact sets LEE I Zij. Zin, A (1) {0 ee i „ap pN Gi 1 eT vg, | (41) Therefore, all the signals in a closed-loop system are 3 bounded. Specifically, the following can be further proved nZi(t) = 2c, (42) At this point, proving of the theorem 1 is completed.
3. Result analysis The validity of the proposed solution is verified by simu- lation experiment analysis. A time-varying reference trajectory x4 = 0.16sin(2.5t) is selected. In order to deal with the problem of unknown symbols of actuation characteristics, the Nussbaum function NG =e x cos (En) is selected to satisfy the corresponding properties. To overcome the jitter phenome- non, arctan{10) is used to replace sign{). The parameters of the fractional order tracking differentiator based on the hy- perbolic sine function are selected as m= 9, ¢ = 12, ¢; =0.2, diy =2 and d, =6.
In order to facilitate the controller design, the smooth function S(u) is introduced to approximate the unknown actua- tion characteristic, and function parameters are 5 =9, n=-01 and w=02. Controller parameters are selected as k;, =12, ky = 20, ky, =20, 0,=8, b,=02, g,=2, 0,=03 and 0, = 0.3. Initial values of all the variables are set as zero. Furthermore, a single-layer Chebyshev neural network is used, and the basis function of the Chebyshev polynomial is &(ey ey) =[1, e, 2e — 14e3 — 3ey, ey Ze — 1,4ej — 3e,]".
Fig. 5 shows position tracking; Fig. 6 shows velocity tracking; and Figs. 5-6 reveal that the blue solid line and the red dashed lines overlap completely in a short period of time. The simulation results show that the solution has high tracking accuracy and high convergence speed.
Fig. 7 depicts the control input of the fractional order arched MEMS resonator under the unknown actuation characteris- tic formed by saturation and deadzone characteristics. It should be noted that the controller can effectively avoid the oscillation of the control input within the asymmetric dead- zone threshold.
Under the condition that no measure is taken, the uncertain direction of the actuation characteristic may cause deteriora- tion of the performance of the controller. Fig. 8 reveals that three curves of the Nussbaum function are consistent at dif- ferent actuation amplitudes. It can be concluded that the problem of the uncertain direction of the actuation character- istic is well solved herein. Fig. 9 shows an intermediate variable Z‚(f).i=L2 having an integral term under different actu- ation amplitudes. All the curves overlap and converge to the neighborhood of zero, which also verifies that the proposed solution has good parameter perturbation capability. Fig. 9 (a) shows an intermediate variable Z(t).i=1 under different actua- tion amplitudes. Fig. 9(b} shows an intermediate variable Z(1).i=2 under different actuation amplitudes.
Considering harsher conditions such as actuation character- istics, chaotic oscillation and model uncertainty, although the actuation amplitudes and the fractional orders of the sys- tem are different due to the uncertainty of the model, Fig. 10 shows that there is no vibration phenomenon within the asym- metric deadzone threshold in the actual control. In addition, even if the entire system is subjected to harsh conditions,
the control input achieves no flutter. Fig. 10 shows control input under different actuation amplitudes and fractional or- ders; (a) shows control input under different actuation amplitudes; (kb) shows control input under different fractional orders; Fig. 11 shows updating single weights under different actu- ation amplitudes and fractional orders; (a) shows updating single weight under different actuation amplitudes; (b) shows updating single weight under different fractional orders; The updating single weight 7,(1) of the Chebyshev neural network di- rectly affects the approximation performance of the high-order nonlinear function. The fractional calculus has dynamic model- ing and nonlinear control capabilities, and can more accurately describe the dynamic characteristics of the system.
However, the fractional order value may cause chaotic oscilla- tion and controller flutter. Fig. 11 shows that chaotic oscillation and controller flutter are well solved herein.
Fig. 12 is a phase diagram under different actuation ampli- tudes and fractional orders; (a) is a phase diagram under different actuation amplitudes; (b) is a phase diagram under different fractional orders; Compared with Figs. 3-4, Fig. 12 reveals that the fractional order arched MEMS resonator switches to a regular motion state and the inherent chaotic oscillation is completely suppressed. The results of Fig. 13 illustrate that the intermediate variable curve is not sensi- tive to changes in the fractional order.
Fig. 14 is a tracking state 2,2 under different actuation amplitudes and fractional orders; (a) shows anti-oscillation performance under different actuation amplitudes; (b) shows anti-oscillation performance under different fractional or- ders. The hyperbolic sine function is regarded as an ideal choice for the tracking differentiator. That is because sinh(}
becomes linear when close to 0, and sinh(} becomes nonlinear when away from 0. On one hand, the nonlinear characteristic can converge quickly, and the linear characteristic can elimi- nate flutter. On the other hand, it is very difficult to directly solve ‚„3/a,(f) subjected to complex calculation. Fig. 14 shows that the tracking differentiator has excellent track- ing function approximation performance and anti-oscillation performance for different actuation amplitudes and fractional orders.
To further illustrate the advantages by using the solution of the tracking differentiator (HSF) based on the hyperbolic sine function, two solutions, i.e., a tracking differentiator (NTD) and a sliding mode based tracking differentiator (SMTD), are introduced for comparison.
Fig. 15 shows that the SMTD has minimal flutter within 0.5 second and the control inputs of the three solutions are sub- stantially equal after 0.5 second. However, Figs. 16-17 show that the inventive solution (blue line} is significantly bet- ter than NTD (red dashed line) and SMTD (green dashed line) due to minimal jitter and amplitude.
Finally, it should be noted that the above preferred embod- iments are only used for describing, rather than limiting the technical solution of the present invention. Although the pre- sent invention is already described in detail through the above preferred embodiments, those skilled in the art shall understand that various changes in form and detail can be made to the present invention without departing from the scope de- fined by claims of the present invention.

权利要求:
Claims (3)
[1]
An anti-oscillation adaptive control method for a fractional order vaulted MEMS resonator, comprising the following steps: S1: creating a dynamic model of a fractional order vaulted MEMS resonator with an unknown actuation characteristic using the Galerkin decomposition method ; and S2: designing an adaptive driver.
[2]
The anti-oscillation adaptive control method for the fractional order domed MEMS resonator according to claim 1, wherein the step S1 is specific as follows: the dynamic model of the fractional order domed MEMS resonator having an unknown actuation characteristic is written as follows by use to make up the Galerkin dissolution process: '. LoL Ae costa) <4 ~ ax is} x == Lesson jx: {7} + ee RARE ij i vn * * RR og Hia FIRS CT | gtd el) where variables are defined as follows: "Vu represents a ratio; £ = 2/5, fm, represents a dimensionless time variable; g = Cm, fb, represents a damping coefficient; Los RES Sn wtih. y = 8.57, by represents a stretching parameter; 2g, ë & 7, represents a voltage parameter, coy = & 4 '%; / 5 ,, represents frequency, & = A3; represents an initial rise, “ist; represents a constant; w = gin dic charge: x Sv ha ij, roordi Iity: «xy = git} represents displacement,“ represents velocity, represents a fractional order, £ represents a symbol defined by Caputo in the field of fractional computation, AM {2} represents an unknown ~ A. Co FET ... actuation characteristic; ZL represents a longitudinal coordinate ;, @ represents a first normalized mode shape; * £, represents deflection; = 24, represents dimensionless quantity; To gps representwo orders time; € JE ufo CRE) = gels), as i A {x} HT i. {a ex represent constants; i: represents current Be, Gel, gt control input; & & ‚K> 0, L represents length; A represents a section area; & represents width, £, represents a viscous damping coefficient; & represents thickness; È represents Young's modulus; I. represents moment of inertia; 8 represents mass density; # 2, represents harmonic load frequency; se, represents the permittivity of vacuum; ¥. represents DC voltage, - represents AC voltage, wy, represents vaulted displacement, «:, represents actuation frequency; where an asymmetric non-smooth saturated non-linear actuation characteristic {wu} exists in the Input and is expressed as
UA TE | {ia it {11} aftisus Ho Mads | f, AND {ry aca {) x 8 bgt {1} “4, SRE lt} | i, SE 4 a) (2) where # and x represent limits; «, {F} and = {tf} represent time-varying functions, +, and I, represent functions of dead zone characteristics; it. and 249, represent positive unknown breakpoints; because a, {t} and a. {t} are time-varying, asymmetric, saturated, and non-smooth, an equalization function is introduced for the approximation of an asymmetric non-smooth saturated property Mu) = Si) + D1) (3 ) and
L.A SH emu Su} TT Wise Wu (4) where ww represents a design parameter, {wu} represents an approximation error, and | D {u} l = Mia) - Sta) = I "; where I" represents a positive and unknown constant; according to mean value theory holds for the equalization function {i}:: ae grte) ee = SU) a again Sti) ZT an u (5) by defining »= 85 {1} / 8u and obtaining 5 {85 = 9, (3) is rewritten as: Mu) = pu + Dia) (6) 0 a {t} SR. CL mi i (2) and isle) represent time-varying functions and can represent the current state of a non-linear system under internal and external interference; such equalization functions have only upper and lower limits of the actuation characteristic. . . “Orst Ww orresult in different needed, different values for approximation coefficient approximation results for Af {z}.
system parameters are set as y = 7.993, A = 03, u = 01, e = 0935, £ = 119.9883 and ox = 0.4704; the chaotic oscillation of the fractional order vaulted MEMS resonator is revealed by different fractional orders and drive amplitudes using a variable step size START / TRBDF2 solver; transient chaos occurs at fractional order values such as 2 = 1.8 and © .35; then the domed MEMS resonator suddenly switches to a non-chaotic state at = = 2.3 and 0.75; definition 1: the Caputo definition of f {£} in the fractional derivative is expressed as § ve f ”{x} 5 ee Pils © FUT (7) i and Ë [7} 2 £ 3 = 3 i Fe J where Min — a ) = i stp lg, CON represents gamma function; + and Finlryy represent an integer; ° order derivative of f (y}; lemma 1: for continuous functions # ° {t} and f '{t} the following equation proceeds i Sj 3 is $ y 3 1 SFE oa oF Fg vid PE a J: a} JE) hl {t} ~ J {ed sad VE) CEOS AEE 3 ee) | BALI ds | ir af THe) i diy (7) Pima) ees) {te} © {eer} (8) where © <0 a <1; the lemma 1 and a relational expression 1 | os dhl) I “0b can be used ite} AND fr-eiimE] 3 the obtain: to obtain: nd 42 = pK od fF g eG (1) 2 / (1). Gor (1) <0 ©) Loo EEE - Eafe - Ora where PATA it) = Iz {te} a and Ga wr 1; F {t} E RE and lemma 2: the fractional order system%; 1 (¢) = 0 (1) Beda di; files git) ex are converted to a linear continuous frequency distribution model of a fractional order integrator; FEE Let} 5 Lots i = —e ¥ (a) + O1), i IN and - - WII}, u LenS on, Odie, (10) where u Tes = sielwe: represents a weighting function, P {a, t} zE represents a current state of the system; definition 2: if the function &{#} meets the following properties iis sum 1 iF Nig) an = hen ‚Yom, sup dy Nln) dn = + (1) fam OT EZ i AF led = —on Yom inf | Napa 20 (12)
the function is called a Nussbaum function; the Nussbaum function is considered an effective means of dealing with an unknown symbolic problem in an actuation characteristic, the next lemma introduced related to the Nussbaum function to support driver design and stability analysis;
ve) al) lemma 3: it is assumed that and are in [0 =), that there is a v (r) so Nt) equalization function for it, and that is a Nussbaum function; then the next inequality goes to Pisae ”| | zi Nine ndr re | andr 38 (3) where = ©; git} is a non-zero constant; g, represents a clean constant, then V {&}, nit) and [7 g {7) NGI} DF ndt are bound; assumption 1: a reference course x; and n ° order derivative are known and limited, while state variables x, {t} and x. {t} can be measured in the meantime; for the fractional order domed MEMS resonator with uncertainty and time-varying actuation characteristic, an adaptive control solution is proposed so that output ¥ = x; {£} follows the reference curve x, with a small error; and at the same time, oscillation related to chaotic behavior and asymmetric dead zone is completely suppressed.
[3]
The anti-oscillation adaptive control method for the fractional order domed MEMS resonator according to claim 1, wherein the step S2 is specific as follows: a Chebyshev polynomial is selected in the form of two repetition formulas I, (XY) = 23 (X ) -7_, (XY). I, (x) = 1 (14) where ve 5 and TAX) are defined as ¥, 25, 23x - 1 or 2x + 1, for [rx] ER ”, an amplified form of the Chebyshev polynomial is constructed as: where Ti (xj) i = 1 ... nj = 1 ... m represents the Chebyshev polynomial, & (X) represents a basic function vector of the Chebyshev polynomial, n represents the order; for any given unknown continuous function f (X) on a compact set, YES) is approximated with complete accuracy to the continuous function based on a general Chebyshev neural network approximation theory to obtain where p (t) is an equal weighting vector; there is a Chebyshev neural network Xan;
where & {X}> © is an approximation error; 24 and Dg; clean boundary represent compact sets of {tf} and JX, respectively; an optimal parameter & * is set to be equal to nrg min | Sup IF (fl) |
* LISBy Bit) = el) - ¢ * {£), where ¢ * is mentioned as artificial quantity, if Ex »0 then Jel} zE;
to promote fast online computation, the following conversion is used to reduce the number of weighting vectors of the Chebyshev neural network go L (A) rd IJE IX Eat sed, (18) where 7 x7 3; the relational expression A (t} = A, {t} - this) holds; 5 A is an estimated value of &, {t}; & is a small positive constant; 5 a mathematical transformation of the Chebyshev neural network associated with the number of weighting vectors is derived by means of Young's inequality; step 1: defining a first intermediate variable LF IFAS (19) where the tracking error e, {¢} is defined as e, {t} = x, {¢} —x, {t}; =, represents a positive design parameter; if Z,; {t} = © and wt - on, then e, {t} 39, fe {tydr - 0, Vtm a second intermediate variable e, {t} = x. {t} - «, {t} where the tracking error Et = et) + oy i e.it) art is selected, where%: = © is a design parameter and «{£) represents virtual control, a derivative of Z, {t} is derived from Caputo's definition fractional calculus; it {{| = 7} Pill U} “I 6 [§ dT vag, { } = e SLX {1}) (20) virtual control is selected if:
cir) = k {thre | e {dr -oe {hee Sif, STF STR Jg dt 7 ne Le MEE YF (21) where k12 represents> 0 control gain; based on lemma 1, the following continuous frequency distribution model is obtained: CO {ant} a: PARE Eg. S NF Say a fa. Ys NY Fe SRE fF RE zn wi, Ie {3 a dt i: + Ey {ice 7, | € i i id oh Si {§} we cl {£ i, $ d 8 | oy {i i 2e i Hoy fa} P {Wd, X} thick (22) Lyapunov stability criteria are considered (9 + 1st egg) the § F 3 g Tm ATES fA (23) sa) seeding} where “= far)
a derivative over time of Vi (t) is calculated as. Ar. . . -. . N SEINE hu Cs and Zi gi LR sa TV fe) Cg 5; {} Re = | with, {i} ¥, {av flaw kd i {{} + ha 8} ay if} (24) step 2: selecting fractional order Lyapunov stability criteria Fav ~ la Hy Fa Js {ad ‚31 a 2 do # EN ie, x dy jes i has PA & 2 (25) at & = 0 aq. fad = sini}. fio) since] where & = 8, le} = and gy fw} = "3" -
Z> (t) is derived to obtain CEE AND SE TN ee Feb © FL bs des {i} wv Fie € “ig {Es {#} Ty Ely is} + Yes {} (26)…. Lo Ae dg coste th CL == oe. At} “ls vine) AHI Bp costes 1) vrank li} ma edig}. Za Hed x (0) where Bago); fi} is a high order nonlinear function, where system parameters such as &, §, + and &; ,, cannot be accurately measured, and it is very difficult to construct an accurate system model due to the influence of internal and external factors ; various external actuations generate damaging oscillation for the domed MEMS resonator, and such oscillation decreases the efficiency of the system by a certain amount, to solve these problems, Chebyshev neural network is used (rE CY AND JL Alsmeer) | == (27) in reality Za, (r) cannot be solved directly due to the complexity of the computation; to solve this problem, a fractional order tracking differentiator is designed based on a hyperbolic sine function to estimate the fractional order derivative of the virtual controller «. {t}; 8 _ Je 4422) = Zan and 2. N NE 4 Ie Gz, == [e sinh (d, (2. —d, (1) + €, sinh (dz,) / | (28)) where the states z. And,% 4, (t) of the tracking differentiator based on the hyperbolic sine function are equal; 2 20, £> 08 {= 123 end, =, {= 12 are design constants; a relational expression fim i ES | - ait) | dt = 8 with Emma ve = n positive T exists; (27) and (28) are exchanged in (26) to obtain SE EE Fp) AE Ol te eo)} sz om with) (29) EE a) EE Fed SEER ij BITE .. ee SVBSRL JS VVT can be easily distracted; a continuous frequency distribution model is further derived using lemma 1; Te or ON VO, fan dd ee re ea oy REE MRNA NY.
TEPER AND URINE il ellievas ì de A ad ax SWNT od ARS Kee SO AJ LA = | ge Lab, Leth du, ij EH ARE, OW, Let] OE wf thou, Ti Eb ZEN, {2} (30) the derivative in time of (25) is taken. Py ALE fee Hs ie} AND: Sieve TT EE tee git i {ei FAs dk wi le} Ltda + Ls i =; =: i | gt Zels) i | wity {ud V, te EN {1} EN {4 JE {2,2 JE {2.00 i Legs Be &, To it, 28 | C2 31)
at which ; the Nussbaum function is used to set up the following control input Lo ann once lee): A and ka Nl 0H wh, EE EE a, EHH where &, 228 and A> 0 are control gains, and the following update law has CoE e = Ak FE i TIA 3 is a ee $ 0 3 2 5 Ey May + = & “ax! > 3. Sins Zl ok Ze Ne oo lk cigaZ ien vol. (34) where g, is a positive number; the update law and control law are interchanged in (31) to obtain {tf} § 2 pe Cn. i Wik, {we} Wl [aa Hw es. and | "(35)" fay is) Ale ad in) vague} pe theorem 1: a fractional order vaulted MEMS resonator with unknown actuation characteristic is considered assumption 1 holds, if the anti-oscillation adaptive control method (32) composed of the adaptive speeds (33) and (34) are introduced, then all internal signals are kept limited and oscillation including the chaotic behavior and the asymmetric dead zone is completely eliminated; proving: the entire Lyapunov candidate position is defined as cu 1 Se ea Cn hs i a EE 3 216 Fom | u oe ddie en} 4 Joa Vi feo file (36) 3 2] 2 He {eo} &: li €] de + 2% da Ha, (u} + {od ils Zora 2 ae oia Û "eo according to Gado OLD 2 2g 30 + Lp ai ey the following is obtained
I wen es lu: 5 a pw yo Bo MARNE ARS RL Rd =; : ayes ~ 5 “i rr {i EES Fos defo oe ET oe Ei 72 ae ay A 3 Fria 2e i op, Le bedjes be lpt] sg A ki} 1, Ze) + EY 37) Se plaNlg} lg At Mk Ai) ey where I amen. BE ee by £ ‚= mini, Ik, 0.) 1e, the above formula is simplified to ee EE 28 Fis oF {se aim {aN {n + 1 Fy (38) both sides of the above formula are multiplied by s7: t to obtain the following
I gE ie ”} Co Lo u 39 EE OE IE Ne sce (39) 2 = ss Le & Ë = Eg by | _%, the above formula is integrated 2x 5 OsrljseFiljse ”{cin pN {n} + d jetide {es} by (40) Zt), Zl!) and A. {£} belong to compact sets Gli SEO a cal (p8 {ne ede vg, “4h pi mY therefore, all signals are limited in a closed loop system; the following is further proven EmZiie) <2c, (42) here 1s completes the proof of theorem 1.

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引用文献:

---

公开号 | 申请日 | 公开日 | 申请人 | 专利标题

---

CN106647277A|2017-01-06|2017-05-10|淮阴工学院|Self-adaptive dynamic surface control method of arc-shaped mini-sized electromechanical chaotic system|

---

US7812680B1|2005-05-03|2010-10-12|Discera, Inc.|MEMS resonator-based signal modulation|

---

CN105204343B|2015-10-13|2018-05-15|淮阴工学院|The Nano electro-mechanical system backstepping control methods inputted with output constraint and dead band|

---

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---

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---

CN110299141B|2019-07-04|2021-07-13|苏州大学|Acoustic feature extraction method for detecting playback attack of sound record in voiceprint recognition|

---

CN110286595A|2019-08-12|2019-09-27|金陵科技学院|The new fractional-order system self-adaptation control method that one kind is influenced by saturation nonlinearity input|

---

CN110568759A|2019-09-26|2019-12-13|南京理工大学|robust synchronization control method of fractional order chaotic system|

---

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申请号 | 申请日 | 专利标题

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CN201811543428.5A|CN109613826B|2018-12-17|2018-12-17|Anti-oscillation self-adaptive control method of fractional-order arched MEMS resonator|

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