·¢²¼Ê±¼ä : ÐÇÆÚÒ» ÎÄÕ»ùÓÚMatlabµÄÐźÅÓëϵͳʵÑéÖ¸µ¼ - ͼÎĸüÐÂÍê±Ï¿ªÊ¼ÔĶÁ88b5f2423169a4517623a351
¹ýsymº¯ÊýÀ´¶¨Òå¡£
?tÀý1£ºÓÃMATLABµÄlaplaceº¯ÊýÇóf(t)?esin(at)u(t)µÄFT¡£
½â£ºMATLABµÄÔ´³ÌÐòΪ£º >>f=sym(¡®exp(-t)*sin(a*t)¡¯); >>L=laplace(f) »ò
>>syms a t
>>L=laplace(exp(-t)*sin(a*t));
(f,v)¡£Ëü·µ»ØµÄº¯ÊýLÊǹØÓÚ·ûºÅ¶Ôlaplaceº¯ÊýÁíÒ»ÖÖÓï¾ä¸ñʽΪ£ºL?laplaceÏóvµÄº¯Êý£¬¶ø²»ÊÇĬÈϵÄs¡£¶ÔÉÏÀýÖÐÈç¹ûÒªÇóFTºóµÄ±í´ïʽ×Ô±äÁ¿Îªv£¬ÔòMATLABÔ´³ÌÐòΪ£º
>>syms a t v >>f=exp(-t)*sin(a*t); >>L=laplace(f,v)
[×¢£ºÇë×ÔÐÐÑéÖ¤½á¹ûÕýÈ·Óë·ñ] £¨¶þ£©ÀÆÕÀ˹·´±ä»»£¨ILT£©
1¡¢»ùÓÚMATLAB·ûºÅÊýѧ¹¤¾ßÏäʵÏÖILT
Èç¹ûÁ¬ÐøÐźÅf(t)¿ÉÓ÷ûºÅ±í´ïʽ±íʾ£¬Ôò¿ÉÓÃMATLABµÄ·ûºÅÊýѧ¹¤¾ßÏäÖеÄ
(L)¡£ ilaplaceº¯ÊýÀ´ÊµÏÖÆäILT£¬ÆäÓï¾ä¸ñʽΪ£ºf?ilaplaceʽÖÐf·µ»ØµÄÊÇĬÈÏ·ûºÅΪ×Ô±äÁ¿tµÄ·ûºÅ±í´ïʽ£¬LÔòΪsÓò·ûºÅ±í´ïʽ£¬Ò²¿Éͨ¹ýsymº¯ÊýÀ´¶¨Òå¡£
s2F(s)?2s?1µÄILT¡£ Àý2£ºÊÔÓÃMATLABµÄilaplaceº¯ÊýÇó
½â£ºMATLABÔ´³ÌÐòΪ£º
>>F=sym(¡®s^2/(s^2+1)¡¯); >>ft=ilaplace(F) »ò
>>syms s
>>ft=ilaplace(s^2/(s^2+1)) [×¢£ºÇë×ÔÐÐÑéÖ¤½á¹ûÕýÈ·Óë·ñ]
2¡¢»ùÓÚMATLAB²¿·Ö·Öʽչ¿ª·¨ÊµÏÖILT
ÓÃMATLABº¯Êýresidue¿ÉµÃµ½¸´ÔÓÓÐÀíʽF(s)µÄ²¿·Ö·Öʽչ¿ªÊ½£¬ÆäÓï¾ä¸ñʽΪ£º[r,p,k]?residue(B,A)
ÆäÖÐB¡¢A·Ö±ð±íʾF(s)µÄ·Ö×ӺͷÖĸ¶àÏîʽµÄϵÊýÏòÁ¿£»rΪ²¿·Ö·ÖʽµÄϵÊý£»pΪ¼«µã£»kΪF(s)ÖÐÕûʽ²¿·ÖµÄϵÊý¡£ÈôF(s)ΪÓÐÀíÕæ·Öʽ£¬ÔòkΪ0¡£
F(s)?s?2s3?4s2?3sµÄILT¡£
Àý3£ºÀûÓÃMATLAB²¿·Ö·Öʽչ¿ª·¨Çó½â£ºMATLABÔ´³ÌÐòΪ£º >>format rat; >>B=[1,2]; >>A=[1,4,3,0]; >>[r,p]=residue(B,A)
³ÌÐòÖеÄformat ratÊǽ«½á¹ûÊý¾ÝÒÔ·ÖÊýµÄÐÎʽ±íʾ£¬ÆäÔËÐнá¹ûΪ£º r= -1/6
-1/2 2/3 p=
-3
-1 0
´ÓÉÏÊö½á¹û¿ÉÖª£¬F(s)ÓÐ3¸öµ¥Êµ¼«µã£¬¼´p1??3,p2??1,p3?0£¬Æä¶ÔÓ¦²¿·Ö·Ö
F(s)?2/3?1/2?1/6??ss?1s?3¡£Ëù
ʽչ¿ªÏµÊýΪ£º-1/6¡¢-1/2¡¢2/3¡£Òò´Ë£¬F(s)¿ÉÕ¹¿ªÎª£º
1?21?f(t)???e?t?e?3t?,(t?0?)6?32?ÒÔ£¬F(s)µÄ·´±ä»»Îª£º
F(s)?s?2s(s?1)3µÄILT¡£
Àý4£ºÀûÓÃMATLAB²¿·Ö·Öʽչ¿ª·¨Çó
½â£ºF(s)µÄ·Öĸ²»ÊDZê×¼µÄ¶àÏîʽÐÎʽ£¬¿ÉÀûÓÃMATLABµÄconvº¯Êý½«Òò×ÓÏà³ËµÄÐÎʽת»»Îª¶àÏîʽµÄÐÎʽ£¬ÆäMATLABÔ´³ÌÐòΪ£º
>>B=[1,-2];
>>A=conv(conv([1,0],[1,1]),conv([1,1,[1,1]])); >>[r,p]=residue(B,A) ³ÌÐòÔËÐнá¹û£¨ÂÔ£©
F(s)?223?2???s?1(s?1)2(s?1)3s
¸ù¾Ý³ÌÐòÔËÐнá¹û£¬F(s )¿ÉÕ¹¿ªÎª£º
?t?t2?tËùÒÔ£¬F(s)µÄILTΪ£ºf(t)?(2e?2te?1.5te?2)u(t)
£¨Èý£©ÀÆÕÀ˹±ä»»·¨Çó½â΢·Ö·½³Ì
ÀÆÕÀ˹±ä»»·¨ÊÇ·ÖÎöÁ¬ÐøLTIϵͳµÄÖØÒªÊֶΡ£LT½«Ê±ÓòÖеij£ÏµÊýÏßÐÔ΢·Ö·½³Ì£¬±ä»»Îª¸´ÆµÓòÖеÄÏßÐÔ´úÊý·½³Ì£¬¶øÇÒϵͳµÄÆðʼÌõ¼þͬʱÌåÏÖÔڸôúÊý·½³ÌÖУ¬Òò¶ø´ó´ó¼ò»¯ÁË΢·Ö·½³ÌµÄÇó½â¡£½èÖúMATLAB·ûºÅÊýѧ¹¤¾ßÏäʵÏÖÀÆÕÀ˹Õý·´±ä»»µÄ·½·¨¿ÉÒÔÇó½â΢·Ö·½³Ì£¬¼´ÇóµÃϵͳµÄÍêÈ«ÏìÓ¦¡£
Àý5£ºÒÑ֪ijÁ¬ÐøLTIϵͳµÄ΢·Ö·½³ÌΪ£ºy??(t)?3y?(t)?2y(t)?x(t)£¬ÇÒÒÑÖª¼¤
?2tÀøÐźÅx(t)?4eu(t)£¬ÆðʼÌõ¼þΪy(0?)?3,y?(0?)?4£¬ÇóϵͳµÄÁãÊäÈëÏìÓ¦¡¢Áã×´
̬ÏìÓ¦ºÍÈ«ÏìÓ¦¡£
½â£º¶ÔÔ·½³ÌÁ½±ß½øÐÐÀÆÕÀ˹±ä»»£¬²¢ÀûÓÃÆðʼÌõ¼þ£¬µÃ£º
s2Y(s)?sy(0?)?y?(0?)?3[sY(s)?y(0?)]?2Y(s)?X(s) ½«ÆðʼÌõ¼þ¼°¼¤Àø±ä»»´úÈëÕûÀí¿ÉµÃ£º
Y(s)?3s?13X(s)?s2?3s?2s2?3s?2
ÆäÖУ¬µÚÒ»ÏîΪÁãÊäÈëÏìÓ¦µÄÀÆÕÀ˹±ä»»£¬µÚ¶þÏîΪÁã״̬ÏìÓ¦µÄÀÆÕÀ˹±ä»»¡£ÀûÓÃMATLABÇóÆäʱÓò½â£¬Ô´³ÌÐòÈçÏ£º
>>syms t s
>>Yzis=(3*s+13)/(s^2+3*s+2); >>yzi=ilaplace(Yzis) yzi=
-7*exp(-2*t)+10*exp(t) >>xt=4*exp(-2*t)*Heaviside(t); >>Xs=laplace(xt); >>Yzss=Xs/(s^2+3*s+2); >>yzs=ilaplace(Yzss) yzs=
4*(-1-t)*exp(-2*t)+4*exp(-t) >>yt=simplify(yzi+yzs) yt=
-11*exp(-2*t)+14*exp(-t)-4*t*exp(-2*t)
?t?2ty(t)?(10e?7e)u(t) ziϵͳµÄÁãÊäÈëÏìӦΪ£º
?t?t?2ty(t)?(4e?4te?4e)u(t) ϵͳµÄÁã״̬ÏìӦΪ£ºzs?t?2t?2tϵͳµÄÍêÈ«ÏìӦΪ£ºy(t)?yzi(t)?yzs(t)?(14e?4te?11e)u(t)