Approximating the Time Optimal Solution for Accessible Unknown Systems by Learning using a Virtual Output

Time optimal control of systems with bounded inputs is a numerically awkward problem as essentially a whole trajectory has to be designed on the basis of a single final point at unknown time. Analytical solutions are possible only for very few problems, in general numerical techniques will be needed, which, in view of the non convexity of the associated optimization problem, will typically converge to a local minimum. In case of an unknown but experimentally accessible nonlinear system, such a solution cannot be found numerically, but a learning algorithm has been proposed which typically converges to a solution not far from the optimal one. However, as in the case of numerical computations, no guarantee of global optimality can be given. In the numerical case, different randomization techniques can be used to ascertain the existence of better solutions, e.g. by choosing different initial conditions. This is more difficult in the case of an experimental method for unknown systems, but this paper proposes an approach based on virtual outputs (linear combination of measurable states) for the same goal which is shown to work in a classical problem of counterintuitive time optimal control - the fastest climbing of a plane.