Plotting and Evaluating PROPT Results
After solving a problem with PROPT, you will probably want to evaluate and/or plot the results.
In PROPT, state and control variables (tomStates and tomControls) are functions of a set of coefficients, and of time. When numeric values are given for the coefficients and for time, states and controls get numeric values.
Example
For example, let's create a couple of states:
>> toms t >> setPhase(tomPhase('p',t,0,2*pi,10)) >> tomStates x y
We can see how x is constructed by converting it into m-code:
>> [code, tempD] = mcode(x) code = out = interp1p(tempD{1},x_p,t); tempD = [11x1 double]
So x depends on time t and on the coefficient vector x_p. When solving for x in a set of ODE:s or DAE:s, we are actually solving for the coefficients x_p.
For example, we can solve a simple set of ODE:s like this:
>> solution = ezsolve(0,{collocate({dot(x)==y,dot(y)==-x}),initial({x==1,y==0})})
solution = x_p: [11x1 double] y_p: [11x1 double]
(Note that the solution does not contain values for t. That is because t is not an unknown t.)
Once we have a solution, we can substitute that into any symbolic expression. For example:
>> x_sol = subs(x,solution) >> mcode(x_sol) ans = out = interp1p(tempD{2},tempD{1},t);
Now, x_sol is a symbolic expression that only depends on t. If we substitute in a numeric value for t, then we get a numeric result:
>> subs(x_sol,t==0.2) ans = 0.9801
It is possible to substitute in a vector of values, one at a time, using atPoints
>> atPoints(1:6,x_sol)
ans = 0.1153 -0.8754 -1.0613 -0.2714 0.7680 1.1013
Some common substitutions have shortcuts, for example initial(x_sol) sets t to the inital point on the active tomPhase, and collocate(x_sol) uses the collocation points.
The command collocate(t) returns a list of the collocation points.
In order to plot a tomState, one can of course use atPoints to create a vector of values to plot, but there is also a short-cut, called ezplot. As soon as the solution has been computed we can do:
>> ezplot([x,y])