function?eg1OC
%EG1OC????Example?1?of?optimal?control?tutorial.
%????This?example?is?from?D.E.Kirk‘s?Optimal?control?theory:?an?introduction
%????example?5.1-1?on?page?198?-?202

%?State?equations
syms?x1?x2?p1?p2?u;
Dx1?=?x2;
Dx2?=?-x2?+?u;

%?Cost?function?inside?the?integral
syms?g;
g?=?0.5*u^2;

%?Hamiltonian
syms?p1?p2?H;
H?=?g?+?p1*Dx1?+?p2*Dx2;

%?Costate?equations
Dp1?=?-diff（Hx1）;
Dp2?=?-diff（Hx2）;

%?solve?for?control?u
du?=?diff（Hu）;
sol_u?=?solve（du?‘u‘）;

%?Substitute?u?to?state?equations
Dx2?=?subs（Dx2?u?sol_u）;

%?convert?symbolic?objects?to?strings?for?using?‘dsolve‘
eq1?=?strcat（‘Dx1=‘char（Dx1））;
eq2?=?strcat（‘Dx2=‘char（Dx2））;
eq3?=?strcat（‘Dp1=‘char（Dp1））;
eq4?=?strcat（‘Dp2=‘char（Dp2））;

sol_h?=?dsolve（eq1eq2eq3eq4）;

%%?use?boundary?conditions?to?determine?the?coefficients
%????case?a:?（a）?x1（0）=x2（0）=0;?x1（2）?=?5;?x2（2）?=?2;
conA1?=?‘x1（0）?=?0‘;
conA2?=?‘x2（0）?=?0‘;
conA3?=?‘x1（2）?=?5‘;
conA4?=?‘x2（2）?=?2‘;
sol_a?=?dsolve（eq1eq2eq3eq4conA1conA2conA3conA4）;

%?Compare?the?solutions?from?book?and?Matlab
sol_book?=?{@（t）（7.289*t-6.103+6.696*exp（-t）-0.593*exp（t））...
????????????@（t）（7.289-6.696*exp（-t）-0.593*exp（t））};?????????
time?=?linspace（0220）;
s_book?=?[sol_book{1}（time）;?sol_book{2}（time）];

%?plot?both?solutions
figure（1）;
ezplot（sol_a.x1[0?2]）;?hold?on;
ezplot（sol_a.x2[0?2]）;
ezplot（-sol_a.p2[0?2]）;????%?plot?the?control:?u=-p2
plot（time?s_book‘*‘）;
axis（[0?2?-1.6?7]）;
text（0.60.5‘x_1（t）‘）;
text（0.42.5‘x_2（t）‘）;
text（1.60.5‘u（t）‘）;
xlabel（‘time‘）;
ylabel（‘states‘）;
title（‘Solutions?comparison?（case?a）‘）;
hold?off;
print?-djpeg90?-r300?eg1a.jpg

%%?--------------------------
%????case?b:?（a）?x1（0）=x2（0）=0;?p1（2）?=?x1（2）?-?5;?p2（2）?=?x2（2）?-2;
eq1b?=?char（subs（sol_h.x1‘t‘0））;
eq2b?=?char（subs（sol_h.x2‘t‘0））;
eq3b?=?strcat（char（subs（sol_h.p1‘t‘2））‘=‘char（subs（sol_h.x1‘t‘2））‘-5‘）;
eq4b?=?strcat（char（subs（sol_h.p2‘t‘2））‘=‘char（subs（sol_h.x2‘t‘2））‘-2‘）;

sol_b?=?solve（eq1beq2beq3beq4b）;
%?Substitute?the?coefficients
C1?=?double（sol_b.C1）;
C2?=?double（sol_b.C2）;
C3?=?double（sol_b.C3）;
C4?=?double（sol_b.C4）;
sol_b2?=?struct（‘x1‘{subs（sol_h.x1）}‘x2‘{subs（sol_h.x2）}‘p1‘...
?????????????????{subs（sol_h.p1）}‘p2‘{subs（sol_h.p2）}）;
%?-----------------------------------------------
%?plot?the?result
clear?sol_book?time?s_book;
sol_book?=?{@（t）（2.697*t-2.422+2.560*exp（-t）-0.137*exp（t））...
????????????@（t）（2.697-2.560*exp（-t）-0.137*exp（t））};
time?=?linspace（0220）;
s_book?=?[sol_book{1}（time）;sol_book{2}（time）];

figure（2）;
ezplot（sol_b2.x1[0?2]）;?hold?on;
ezplot（sol_b2.x2[0?2]）;
ezplot（-sol_b2.p2[0?2]）;????%?plot?the?control:?u=-p2
plot（time?s_book‘*‘）;
axis（[0?2?-.5?3]）;
text（1.7‘x_1（t）‘）;
text（0.41‘x_2（t）‘）;
text（.22.5‘u（t）‘）;
xlabel（‘time‘）;
ylabel（‘states‘）;
title（‘Solutions?comparison?（case?b）‘）;
hold?off;
print?-djpeg90?-r300?eg1b.jpg

%%?----------------

?屬性????????????大小?????日期????時間???名稱
-----------?---------??----------?-----??----

?????文件???????4226??2009-11-10?12:54??An?optimal?control?tutorial?for?beginners\eg1OC.m

?????文件???????1614??2009-11-10?17:09??An?optimal?control?tutorial?for?beginners\eg2OC_BVP.m

?????文件???????3414??2009-11-09?03:32??An?optimal?control?tutorial?for?beginners\eg2OC_Descent.m

?????文件???????1133??2009-11-09?20:54??An?optimal?control?tutorial?for?beginners\er3OC_num.m

?????文件???????1113??2009-11-09?13:55??An?optimal?control?tutorial?for?beginners\er3OC_num2.m

?????文件???????1953??2009-11-09?11:34??An?optimal?control?tutorial?for?beginners\er3OC_sym.m

?????文件???????2213??2009-11-09?20:53??An?optimal?control?tutorial?for?beginners\er4OC.m

?????文件????2013854??2009-11-16?00:25??An?optimal?control?tutorial?for?beginners\Solving?optimal?control?problems?with?MATLAB.pdf

?????目錄??????????0??2011-07-06?15:08??An?optimal?control?tutorial?for?beginners

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??????????????2029520????????????????????9