16.( LECTURE非 NUMERICAL5。LT0N 6 F NONLINEAR0 FFEREN升L EQATIONs
NUMERICAL SoL°To ·G川EN舟c。 MPLEX SET of0 YNAMICS 义(t)=斤(x,x) WHERE f()CouL0BE升MNLE升 R FUNCTI0N 工TCNB.工 MOss8 LE To ACTVALLY SoLVE FoR×(t)∈ ACTLY →0 EVELoP A则 UMERICAL SOLUTION. cA小NE0co0 ES TO HELP v5 D。TwsτN MATLAB BUT LET US CONSIDER THE BAsIcs APPROXIMATE THE DEKIUATIVES WITH BACKuARO 0氏 FFERENCES: 义(KT)(KT)-x(k-T T-SMALL FIxED T吨E化ERo0 K- INTEGER工心0EX 义(T)2文Kx)×((K-T) K-2Xk-1+×k
G-2 So,工 F UE WA0 =-3X-4X WJE COULD APP ROXIMATE THIS AS K-+xK-2 3×K-4Xk-xk- Xx-2Xx-I+ XK.-3T2XK-4T(xx-XK-1) 1+4T+3T2 k 2十4T)以k-1 k-2 X K 2+4T 了)Xk-二XK-2 13T2+4T CALLED A RECURSION KELAT0八 GWEN XK-l, XK-1 UE CAN FIND XK THEN USE XK-I Xx To FiND XK+ Mow 00 WE START? 工Fx()=4;×()-3 ()=4月ND×。-x-1=3T X4-3T SIMPLE AIPRo ACH BVT LIMITED ACcVkAcy KEEP T SMALL
C-3 8 16.61 -Numerical example for \ddot x 4 \dot x +3x=0 clear all T=0.05 号 actua1工C 号 start numerl.cs xml=x0-xOdot*T; NN=100; X=[xm1 x0] for li=3+[0: NN X(ii)=((4*+2)*X(ii-1)X(i-2))/(1+4*T+3*T~2) end 4 f );c1f sys=ss([01;-3-4],[01],[10],0 Y=initial(sys, [xo xOdot]',T*[0: NN]') 2.5 120
6-4 IN THE CASE THAT F( K) Is LINEAR, WE cA50凵 E THE E MATLAB USING L OFTEN FWD THAT LINEAR 0YN AMICS COUPLE 0 RE THA小 NE VARIA BLE >CAN ALWAYs WRITE THE Y NAMICS As X=升x+Bu WERE x TS A VECTOR of VARIABLES → CALLED THE STATE ExAMPLE: HILLS E& DATIONs FOR TWO CLsELy SPACEO SPACECRAFT: nw 2/a0 wins X=2 ∧+3n2×+fx ATR升CK 2^￥ RAOJAL LET文 0x+|00 久 2 2A D 0 LINEAR MATRIX FORM THAT CAN BE OIRECTLY s0LE0工 MATLA B