Users Guide for dyn.Opt, Example 5a, 5b, 5c
Stirred-Tank Chemical Reactor - Kirk, D. E., Optimal control theory: An introduction, Prentice-Hall, 1970.
5a - unconstrained with terminal penalty 5b - unconstrained 5c - control constraint
Find u over t in [0; 0.78 ] to minimize
Does not say u^2 in text
5a:
J = | ∫ |
| (x12+x22+0.1*u2)/2 dt + x1(tF)2 + x2(tF)2 |
5b:
J = | ∫ |
| (x12+x22+0.1*u2)/2 dt |
5c:
J = | ∫ |
| (x12+x22)/2 dt |
subject to:
a1 = x1 + 0.25 |
a2 = x2 + 0.5 |
a3 = x1 + 2.0 |
a4 = a2*exp(25.0* |
| ) |
| = −2.0*a1 + a4 − a1*u |
| = 0.5 − x2 − a4 |
x(0) = [0.05 0] |
5b, 5c - x(t_F) = [0 0];
5c - u <= 1
Reference: [16]
toms t for i=1:3
p = tomPhase('p', t, 0, 0.78, 40); setPhase(p); tomStates x1 x2 tomControls u % Initial guess x0 = {icollocate({x1 == 0.05; x2 == 0}) collocate(u == 0)}; % Box constraints cbox = {-1.99 <= icollocate(x1) <= 100 -100 <= icollocate(x2) <= 100 -1000 <= collocate(u) <= 1000}; % x1 cannot be equal to -2, setting to greater % to avoid singularity in a2*exp(25.0*x1/a3) % Boundary constraints cbnd = initial({x1 == 0.05; x2 == 0}); % ODEs and path constraints a1 = x1 + 0.25; a2 = x2 + 0.5; a3 = x1 + 2.0; a4 = a2.*exp(25.0*x1./a3); ceq = collocate({ dot(x1) == -2.0*a1 + a4 - a1.*u dot(x2) == 0.5 - x2 - a4});
options = struct; if i==1 objective = final(x1)^2+final(x2)^2+... integrate((x1.^2+x2.^2+0.1*u.^2)/2); options.name = 'Stirred Tank 5a'; solution1 = ezsolve(objective, {cbox, cbnd, ceq}, x0, options); t1 = subs(collocate(t),solution1); x11 = subs(collocate(x1),solution1); x21 = subs(collocate(x2),solution1); u1 = subs(collocate(u),solution1); elseif i == 2 cbnd = {cbnd; final({x1 == 0; x2 == 0})}; objective = integrate((x1.^2+x2.^2+0.1*u.^2)/2); options.name = 'Stirred Tank 5b'; solution2 = ezsolve(objective, {cbox, cbnd, ceq}, x0, options); else cbnd = {cbnd; final({x1 == 0; x2 == 0})}; cbox = {-1.99 <= icollocate(x1) <= 100 -100 <= icollocate(x2) <= 100 -1 <= collocate(u) <= 1}; objective = integrate((x1.^2+x2.^2)/2); options.name = 'Stirred Tank 5c'; solution3 = ezsolve(objective, {cbox, cbnd, ceq}, x0, options); end
Problem type appears to be: qpcon Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05 ===================================================================================== Problem: --- 1: Stirred Tank 5a f_k 0.014213969120012267 sum(|constr|) 0.000000005238899986 f(x_k) + sum(|constr|) 0.014213974358912253 f(x_0) 0.003474999999999964 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 30 ConJacEv 30 Iter 27 MinorIter 113 CPU time: 0.171875 sec. Elapsed time: 0.172000 sec.
Problem type appears to be: qpcon Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05 ===================================================================================== Problem: --- 1: Stirred Tank 5b f_k 0.016702811155814266 sum(|constr|) 0.000000899223593776 f(x_k) + sum(|constr|) 0.016703710379408040 f(x_0) 0.000974999999999999 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 18 ConJacEv 18 Iter 16 MinorIter 118 CPU time: 0.125000 sec. Elapsed time: 0.125000 sec.
Problem type appears to be: qpcon Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05 ===================================================================================== Problem: --- 1: Stirred Tank 5c f_k 0.000989922252663805 sum(|constr|) 0.000000035597664481 f(x_k) + sum(|constr|) 0.000989957850328286 f(x_0) 0.000974999999999999 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 14 ConJacEv 13 Iter 10 MinorIter 139 CPU time: 0.078125 sec. Elapsed time: 0.078000 sec.
end
subplot(2,1,1) plot(t1,x11,'*-',t1,x21,'*-'); legend('x1','x2'); title('Stirred Tank state variables'); subplot(2,1,2) plot(t1,u1,'+-'); legend('u'); title('Stirred Tank control');