基于滑模的精确差分和滤波

1、滑模控制方法文献学习基于滑模的精确差分和滤波基于滑模的精确差分和滤波鲁棒精确差分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口门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 认 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,实用

7、一阶 RED假设“。由基信号(base signal)和噪音两部分组成，其中基信号的微分存 在Lipschtiz常数。A 0。为求取未知基信号的微分，建立如下辅助方程(0-1)(0-2)X =u.应用2-滑模算法以保证工一/） =0 ,可得到口 =峋一八|工一/（）|“，gn（工一/（）% Ui = - ft sgn（x - /）,1 / 18滑模控制方法文献学习其中d入0为待设计常数，以。为差分器输出。式(0-1)、(0-2)组成系统的解需 结合 Filippov sense求解。定义函数小(4入。=僧俗)|,其中(冲)为以下方程的解:E = - |E|173 + $,(1-|12 + 0,

8、由=(0-3)十0% -园/十出0, =0,卬 coMogc入) 1 ,那么对于有些于巳W(C, 2)将无法实现壮 收敛到，。力3小)越小，收敛速度越快，且当 二网6入=m(7,%1.,如：0时， 参数g，aa值均相同。入增大时，小g,cj)减小。以下为匕收敛到/的 充分条件：口 C 0,入 aOE(q,CA) 0,以。为噪音且|武。|三E ,那么存在常数匕 (与(QC)/*和g+C)/T)使得不等式lud)一外训在有限时间内 实现。假设/,羽为均在离散时间点进行测量，时间间隔为丁，令如士1/分别为连续 的离散测量时刻和当前时刻，即tE民足+ J,那么离散形式的算法为X =(= 一 一 一.：

9、(0-5)i = - ojggnQQJ - /&).2 / 18滑模控制方法文献学习定理3:假设/Q)的微分存在LipscMtz常数C 。，算法(0-5)可在有限时间 暂态后实现u(t) ,其中以为与Q C)/和(a+C)/V有关的常数。1.1.1.3仿真验证仿真初始时刻设置为加=0,输出信号初始值0)=0,利用Euler方法进 行仿真，测量和积分步长均为10，。为与所提出RED进行性能对比，设计线性 差分器为s/(0.U，其即为理想差分器和低通滤波器串联。令口 = 8,人=6 ,当C = 2时，= 0.064,当时，= 0.2,也即对于任何输入信号,，当 其满足了7时，以上参数均可实现精确差

10、分。对输入信号为f(t) =sinE + 5 ,/=siut + 5 -|- O.OlcoalOtj/ft)=菖也古 + 5士十 0.001 cos3保分别进行仿真对 比，结果如下图1-1所示。可见，线性差分器不能实现精确差分，而 RED可实 现高频信号存在时的精确差分。 4 drhfflwikator _ rutl 血口白电 ,2皿皿口1、口闻|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滑模控制方法文献学习4eai drffereniiatex rctust Buuice 哦伯1电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图1-1 RED和线性差分器对比另外，当原始信号存在高频测量噪音时，RED差分精度