UQ eSpace

Abstract

* Multiple-interval, multiple-phase refers to a problem that is modelled using one or more phases and solved on a grid containing one or more intervals. * A nonlinear program is the discrete subproblem that will be solved by the software in place of the continuous time trajectory optimisation formulation. A discussion into nonlinear programming is provided in Section 2.5. Path Constraints ≤ , , (), () ≤ Boundary Conditions The performance index, , has contributions from two terms. The integral term, , is known as the Lagrange cost or the running cost, and the terminal cost term, , is known as the Mayer cost. A combination of the two, as in the above, is known as a Bolza cost (Diehl and Gros, 2017). The decision variables are the initial and final time, as well as the state and control. The constraints are all formulated as inequalities; however, equality constraints can be enforced by setting the upper and lower bounds equal. The above definition has been be expanded to include multiple-phases in Section 4.1. As illustrated in Fig. 2.2, numerous numerical techniques can be implemented to solve a problem that takes the form of Definition 2.1. These numerical techniques are discussed in further detail in Section 2.4. * The active set consists of all constraints that will actually influence the final optimisation. An inequality constraint () ≥ 0 is called inactive at x if (x) > 0. Equality constraints are always active. * The basic idea in homotopy methods is to solve a problem and use its solution as an initial guess for a similar problem. This assists in convergence as it is analogous to warm starting an optimisation.