We will now solve the same problem as in brachistochrone.m, but using a DAE formulation for the mechanics.
In a DAE formulation we don’t need to formulate explicit equations for the time-derivatives of each state. Instead we can, for example, formulate the conservation of energy.
Ekin = |
|
| 2 + |
| 2 | , |
Epot = m g y . |
The boundary conditions are still A = (0,0), B = (10,-3), and an initial speed of zero, so we have
Ekin + Epot = 0 |
For complex mechanical systems, this freedom to choose the most convenient formulation can save a lot of effort in modelling the system. On the other hand, computation times may get longer, because the problem can to become more non-linear and the jacobian less sparse.
Reference: [6]
toms t toms t_f p = tomPhase('p', t, 0, t_f, 20); setPhase(p); tomStates x y % Initial guess x0 = {t_f == 10}; % Box constraints cbox = {0.1 <= t_f <= 100}; % Boundary constraints cbnd = {initial({x == 0; y == 0}) final({x == 10; y == -3})}; % Expressions for kinetic and potential energy m = 1; g = 9.81; Ekin = 0.5*m*(dot(x).^2+dot(y).^2); Epot = m*g*y; v = sqrt(2/m*Ekin); % ODEs and path constraints ceq = collocate(Ekin + Epot == 0); % Objective objective = t_f;
options = struct; options.name = 'Brachistochrone-DAE'; solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options); x = subs(collocate(x),solution); y = subs(collocate(y),solution); v = subs(collocate(v),solution); t = subs(collocate(t),solution);
Problem type appears to be: lpcon Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05 ===================================================================================== Problem: --- 1: Brachistochrone-DAE f_k 1.869963310229847400 sum(|constr|) 0.000000000158881015 f(x_k) + sum(|constr|) 1.869963310388728300 f(x_0) 10.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 168 ConJacEv 168 Iter 93 MinorIter 154 CPU time: 0.203125 sec. Elapsed time: 0.219000 sec.
To obtain the brachistochrone curve, we plot y versus x.
subplot(2,1,1) plot(x, y); title('Brachistochrone, y vs x'); subplot(2,1,2) plot(t, v);