Support variable time with `create_objective_function()`

[moorepants]

We now have variable time step support but it will not work with the current create_objective_function().

[Peter230655]

Would it make sense to give an example how for form the obj_grad function for the cases

• you only want to minimize h_opty
• you want to minimize $\int (fx^2 + fy^2 + hopy) \ dt$ ? I am never 100% sure I do it right, and then if opty does not give a reasonable result I wonder whether my obj_grad is wrong or some other mistake.

[moorepants]

I think the examples are more like $\int dt$ (minimize time) or $w_1\int F^2 dt + w_2\int dt$ (minimize force and time with weights). The integrals should never have 'h' in them, only time. create_objective_function()'s job is to convert integrals in continuous time to their discrete versions.

[tjstienstra]

The simplest would be to use something like the following:

    def lambdify_function(expr, multiplication_array, take_sum):
        if take_sum:
            def integration_function(x):
                return np.sum(x * multiplication_array)
        else:
            def integration_function(x):
                return x * multiplication_array
        return sm.lambdify(
            (states, inputs, params), expr, 
            modules=[{int_placeholder.name: integration_function}, "numpy"],
            cse=True)

    ...

    def multiply_integrals(expr, value):
        if not expr.args:
            return expr
        if isinstance(expr, sm.Integral):
            return value * expr
        return expr.func(*(multiply_integrals(arg, value) for arg in expr.args))

    objective = multiply_integrals(objective, node_time_interval)

[Peter230655]

I think the examples are more like ∫dt (minimize time) or w1∫F2dt+w2∫dt (minimize force and time with weights). The integrals should never have 'h' in them, only time. create_objective_function()'s job is to convert integrals in continuous time to their discrete versions.

Of course STUPID of my part to put h in the integral!