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p A^Œ˘flK MATLAB)£1n ⁄¯‰ ˘&E˘ 11 O ¯Œ˘ Vª1.3\ ¯ SCMATLAB ‚§¿ \demo•-§d Ł X Ø{ . «§S§+ MATLAB 3)Œ˘flK ¡ U {)3MATLABJ«˛e \demo•-§K mXª1-1⁄« I §I L – J « | «SNª1-1MATLAB «§S.¡~X§^r J3-DVisualization «§K Xª1-2⁄« «‘†§ ´ ¥ OpenthisExample98§K Xª1-3⁄« «.¡^r –3T.¡eUU §¯ « ’SN§ ¢yø « Ø 3T§S.¡ e I ¥ 2.b fi Lyapunov §Xe816357492X+X1641931421=123456780`|^lookforlyapunov•- ˛ ’ lyapunovk’ …Œ¶§¿^doc•-… ’…Œ ? N^&E§* ·˜U T § )§¿u ) 211 O ¯Œ˘ Vªª1-2MATLAB «§S.¡ ~ª1-3MATLABN z «§S.¡) Ø /|^MATLAB)ß\ flK§3 ˇ·Øk^ ’X\ ) § Lyapunov §§ \ØX )T § ⁄ §\ AT |¢ e§ww= …Œ –)Lyapunov §ø I lookforlyapunov•-§‰3•-I HelpŁ eU J«|¢’ lyapunov§– Ǒ~§\ U Xe&Edlyap-SolvediscreteLyapunovequations.lyap-Solvecontinuous-timeLyapunovequations.lyap2-Lyapunovequationsolutionusingeigenvaluedecomposition.pdlstab-TestrobuststabilityviaparametricLyapunovfunctions11 O ¯Œ˘ Vª3lyapkr-Lyapunov/Sylvesterequationsolver(Kroneckerproductapproach).dlyapchol-Square-rootsolverfordiscrete-timeLyapunovequations.lyapchol-Square-rootsolverforcontinuous-timeLyapunovequations.l &Ew§12!3^ U • §\ I … A…Œ? &E§X helplyap•-§‰ doclyap•-£ mHTML⁄§K – lyapSolvecontinuous-timeLyapunovequations.X=lyap(A,Q)solvestheLyapunovmatrixequation:A*X+X*A’+Q=0X=lyap(A,B,C)solvestheSylvesterequation:A*X+X*B+C=0X=lyap(A,Q,[],E)solvesthegeneralizedLyapunovequation:A*X*E’+E*X*A’+Q=0whereQissymmetricSeealsolyapchol,dlyap.ReferencepageinHelpbrowserdoclyapØ 'flK§\‹uy ¥ Xe&E 'flK X=lyap(A,B,C)solvestheSylvesterequation:A*X+X*B+C=0¿ \ –Œ A=816357492,B=1641931421,C=−1234567805¿C ˛ ø \ –^e¡ Ø ) 'flK A=[816;357;492];B=[1641;931;421];C=-[123;456;780];X=lyap(A,B,C) (JǑX=0.0748720.089913−0.432920.00807160.48144−0.216030.0195770.182641.1579 (J· · v ' §Q”Ø› !·u Nou Q” { {· (J \ §§ww· · v §A*X+X*B+C (JǑ411 O ¯Œ˘ Vª−1.2212×10−151.3323×10−154.4409×10−165.3291×10−151.7764×10−158.8818×10−167.1054×10−151.7764×10−152.2204×10−15w,§ (J Øp ,§ 0 X … p {))) ) {3.|^د ˇ•-docsymbolic/diff ˛˛ $ ¥ …Œdiff§3 )⁄Jł ˇ&E ˜: `X)~1.1¥ flK§¿ T~’ (J)) 'flKI e¡A ‰1 § †˛ C xsymsx1 § '…ŒL« 5f=sin(x)/(x^2+4*x+3)1n § §’X4 Œ§K – e¡ •-f4=diff(f,x,4)12 MATLAB §S O˜:1.ؘMATLAB ‚§¿ Øtic,A=rand(500);B=inv(A);norm(A*B-eye(500)),toc`$1T ا* (J§¿|^help‰doc•-Ø\ G Ø?1 ˇ&E ˛§¯^ ª§Sª (J )”)øª Ø¥m ' Œ˘„´ ·k)⁄ 500×500 !' ¯Œ A§Ø _§ _ D B§2O AB−I§ ¥IǑ500×500 § Œ¿w« 5§n (JAT·§du^ V Œ ( §⁄–‹ ) Ø•-^u 1øA^ Øs⁄ m2.`^˛ |– “L ı “f(x)=x5+3x4+4x3+2x2+3x+6§¿-x=(s−1)/(s+1)§ f(x)O ⁄s …Œ)C O –ˇLsubs()…Œ¢y§ = (JØ U · { §⁄–I Ø(J?1z{))ø –ˇLsimple()…Œ¢ysymsxs;f=x^5+3*x^4+4*x^3+2*x^2+3*x+6;F=simple(subs(f,x,(s-1)/(s+1))) (Jz{ F(s)=3+23s+54s2+70s3+19s5+23s4(s+1)53.` nŒ√2!6√11!sin1◦!e2!ln(21) 200 k Œi) nŒ –ˇLvpa()…Œ m§ L 5¿§I nŒ= ⁄˛ a.§˜K U ( (JN1=vpa(sqrt(sym(2)),200),N2=vpa(sym(11)^(1/6),200)N3=vpa(sin(sym(pi)/180),200),N4=vpa(exp(sym(2)),200)N5=vpa(log(sym(21)),200) A nŒ200 'OǑN1=1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727350138462309122970249248360558507372126441214970999358314132226659275055927557999505011527820605715612 MATLAB §S O˜:N2=1.4913014754131089242014946687648253445426761253261577981518483147412893600154646617792759695448637729329519548283629181900525497764708600917909528961338083373475281160289205994986450619919284949138885N3=0.017452406437283512819418978516316192472252720307139642683612427640597384203928070042001926791021346914488426873249509483758066561253481112810108146016474143779998382023403544649197667454355161539723988N4=7.3890560989306502272304274605750078131803155705518473240871278225225737960790577633843124850791217947737531612654788661238846036927812733744783922133980777749001228956074107537023913309475506820865818N5=3.0445224377234229965005979803657054342845752874046106401940844835750741559706782346612742534016036157810317232718090066540917564836083066642017599659311781865109872372876561505616330478528283343162475XJ k= §Kk U ⁄ˇ (J§ –g e¡ Ø (JN6=vpa(sin(pi/180),70) (JǑN6=0.017452406437283511653202339175550150685012340545654296875000000000000005¿§ N3 ’§ k16 k Œi· §ø·ˇǑ3 ¡ O Ø¥§k^V Œ ( O pi/180§2^vpa()…ŒØV Œ ?¿ O § , U 3v Œ k Œi⁄–3¡ O Ø¥§ ˘kr w« = Ǒ˛ .C 4.fi Œ˘…Œf(x)=xsinx√x2+2(x+5)§g(x)=tanx§` E …Œf(g(x)) g(f(x)))|^subs()…Œ C O §K – ⁄I E …Œsymsx;f=x*sin(x)/sqrt(x^2+2)/(x+5);g=tan(x);F1=simple(subs(f,x,g))F2=simple(subs(g,x,f)) ª Ø (JǑF1(x)=sin(tanx)tanx(tanx+5)√tan2x+2§F2(x)=tanxsinx√x2+2(x+5)5.duV Œ ( k ‰ Œ § Œ ØJ 3v `^Œ { ˛ $ {O ¿’ C1050§ ¥Cnm=m!n!(m−n)!) n!3MATLABe –ˇLprod(1:n)O §Ǒ –ˇLΓ…ŒO §=dgamma(n+1) §XJn §K˜uV Œ ( Œ ){J– ( )§I /ˇu˛ $ U (/O 5r1=gamma(51)/gamma(11)/gamma(41)m=sym(50);n=sym(10);r2=gamma(m+1)/gamma(n+1)/gamma(m-n+1)12 MATLAB §S O˜:7Œ ) ) { (J'OǑr1=1.027227816999992×1010§r2=102722781706.^MATLAB Ø \ A BA=1234432123413241,B=1+4j2+3j3+2j4+1j4+1j3+2j2+3j1+4j2+3j3+2j4+1j1+4j3+2j2+3j4+1j1+4j¡ ·4×4 §