基于滑模的精確差分和濾波

1、滑?？刂品椒ㄎ墨I(xiàn)學(xué)習(xí)基于滑模的精確差分和濾波基于滑模的精確差分和濾波魯棒精確差分1RED (Robust Exact Differentiation )的局限性Rohusr exact differential ionLet iiipul signals belong (o die space M u, A of measurable functions bounded on a segment鼻 h and let |/| = sup | Define absiruei differentiator as a map associating an output Hgnal with any

2、input signal. A differentiator is called 匕me on seme input if the output coincides with its derivative. The differentiator order is the order of the derivative which it produces. Diflcrentiator D is called robust on seme input f(t) if the output lends uniformly to Df lti while the input signal t?nds

3、 uniformly to/fr). A diffcrenuator is called career on some input if it is 一一act and robust on ill The ideal dillcreatiator cannot be considered as an abstract one, for ir does not operate for n口門di匚 ferentiable inputs.Being exact on two inputs, any differentia I or will actually differendate a diff

4、erence between h心福 inputs which may be considered as a noise. Thus, difierentiator design is a simple trade-ofT the denser the cxiictncss ckiss in 仃，b, the more sensitive will the difereatiator be to small noises. For example, being correct on a thin class of constant inputs, the diflcrcniiator prod

5、ucing ideniiual zero 認(rèn) loially insensitive id nujse. It is easily seen that the correctness wet any abstract diffcrcnTiator uannol be locally deiiiC in liic scl of coiilinuous funcliunti, other* wise small noises with large derivatives would be exactly diffcrcn- tiated, which contradicts robustness,

6、 In particular, no differentiator is correct on all smooth functions or on all polynomials.Lcl 卬(.，升 be the set of all input signals whose n? l)th derivatives have Lipschitzs constant CaQ The statements below Arc valid for any suflicicntly sinall e 0 and noises not exceeding s in itbssuline value,實(shí)用

7、一階 RED假設(shè)“。由基信號(hào)(base signal)和噪音兩部分組成，其中基信號(hào)的微分存 在Lipschtiz常數(shù)。A 0。為求取未知基信號(hào)的微分，建立如下輔助方程(0-1)(0-2)X =u.應(yīng)用2-滑模算法以保證工一/） =0 ,可得到口 =峋一八|工一/（）|“，gn（工一/（）% Ui = - ft sgn（x - /）,1 / 18滑?？刂品椒ㄎ墨I(xiàn)學(xué)習(xí)其中d入0為待設(shè)計(jì)常數(shù)，以。為差分器輸出。式(0-1)、(0-2)組成系統(tǒng)的解需 結(jié)合 Filippov sense求解。定義函數(shù)小(4入。=僧俗)|,其中(沖)為以下方程的解:E = - |E|173 + $,(1-|12 + 0,

8、由=(0-3)十0% -園/十出0, =0,卬 coMogc入) 1 ,那么對于有些于巳W(C, 2)將無法實(shí)現(xiàn)壯 收斂到，。力3小)越小，收斂速度越快，且當(dāng) 二網(wǎng)6入=m(7,%1.,如：0時(shí)， 參數(shù)g，aa值均相同。入增大時(shí)，小g,cj)減小。以下為匕收斂到/的 充分條件：口 C 0,入 aOE(q,CA) 0,以。為噪音且|武。|三E ,那么存在常數(shù)匕 (與(QC)/*和g+C)/T)使得不等式lud)一外訓(xùn)在有限時(shí)間內(nèi) 實(shí)現(xiàn)。假設(shè)/,羽為均在離散時(shí)間點(diǎn)進(jìn)行測量，時(shí)間間隔為丁，令如士1/分別為連續(xù) 的離散測量時(shí)刻和當(dāng)前時(shí)刻，即tE民足+ J,那么離散形式的算法為X =(= 一 一 一.：

9、(0-5)i = - ojggnQQJ - /&).2 / 18滑?？刂品椒ㄎ墨I(xiàn)學(xué)習(xí)定理3:假設(shè)/Q)的微分存在LipscMtz常數(shù)C 。，算法(0-5)可在有限時(shí)間 暫態(tài)后實(shí)現(xiàn)u(t) ,其中以為與Q C)/和(a+C)/V有關(guān)的常數(shù)。1.1.1.3仿真驗(yàn)證仿真初始時(shí)刻設(shè)置為加=0,輸出信號(hào)初始值0)=0,利用Euler方法進(jìn) 行仿真，測量和積分步長均為10，。為與所提出RED進(jìn)行性能對比，設(shè)計(jì)線性 差分器為s/(0.U，其即為理想差分器和低通濾波器串聯(lián)。令口 = 8,人=6 ,當(dāng)C = 2時(shí)，= 0.064,當(dāng)時(shí)，= 0.2,也即對于任何輸入信號(hào),，當(dāng) 其滿足了7時(shí)，以上參數(shù)均可實(shí)現(xiàn)精確差

10、分。對輸入信號(hào)為f(t) =sinE + 5 ,/=siut + 5 -|- O.OlcoalOtj/ft)=菖也古 + 5士十 0.001 cos3保分別進(jìn)行仿真對 比，結(jié)果如下圖1-1所示?？梢姡€性差分器不能實(shí)現(xiàn)精確差分，而 RED可實(shí) 現(xiàn)高頻信號(hào)存在時(shí)的精確差分。 4 drhfflwikator _ rutl 血口白電 ,2皿皿口1、口聞|麻/J , *-I1j-2G24601012eidea II drHerentiatiQini + 1il-crdfir LPF卜T目01口11KIdell diiierBnIialiQni : 2nd-anef LPFJjJ，-411n*4T，.由

11、/ x/ 、/I1 zRotut Exad Diffwemluben. x w3ftMlmstlan wnx3 / 18Robusl DfffBrsntia&Qfi10Idul dirrQntlaAjon + 2nd occter LPF滑?？刂品椒ㄎ墨I(xiàn)學(xué)習(xí)4eai drffereniiatex rctust Buuice 哦伯1電I 印做修idail ditFerenliabon + Isl-crder LPFRobt EkmI DiWruniiatkn10(b)/() =sin+ 5-F 0.01 cos Wtsei differentia!sin + Isl-ordier LPFideal dEFlislsor ruftuil tfifcl dinrwrhHtotf 用后小出H-ifrthk!(C) ,= , 一:, + I L圖1-1 RED和線性差分器對比另外，當(dāng)原始信號(hào)存在高頻測量噪音時(shí)，RED差分精度