Numerical Solution of Optimal Control Problems by Direct Collocation

By an appropriate discretization of control and state variables, a constrained optimal control problem is transformed into a finite dimensional nonlinear program which can be solved by standard SQP-methods [10]. Convergence properties of the discretization are derived. From a solution of this method known as direct collocation, these properties are used to obtain reliable estimates of adjoint variables. In the presence of active state constraints, these estimates can be significantly improved by including the switching structure of the state constraint into the optimization procedure. Two numerical examples are presented. 1 Statement of problems Systems governed by ordinary differential equations arise in many applications as, e. g., in astronautics, aeronautics, robotics, and economics. The task of optimizing these systems leads to the optimal control problems investigated in this paper. The aim is to find a control vector u(t) and the final time tf that minimize the functional J [u, tf ] = Φ(x(tf ), tf) (1) subject to a system of n nonlinear differential equations ẋi(t) = fi(x(t), u(t), t), i = 1, . . . , n, 0 ≤ t ≤ tf , (2) boundary conditions ri(x(0), x(tf ), tf) = 0, i = 1, . . . , k ≤ 2n, (3) and m inequality constraints gi(x(t), u(t), t) ≥ 0, i = 1, . . . , m, 0 ≤ t ≤ tf . (4) Here, the l vector of control variables is denoted by u(t) = (u1(t), . . . , ul(t)) T and the n vector of state variables is denoted by x(t) = (x1(t), . . . , xn(t)) T . The functions Φ : IR → IR, f : IR → IR, r : IR → IR, and g : IR → IR are assumed to be continuously differentiable. The controls ui : [0, tf ] → IR, i = 1, . . . , l, are assumed to be bounded and measureable and tf may be fixed or free. 1 2 Discretization This section briefly recalls the discretization scheme as described in more detail in [18]. Some of the basic ideas of this discretization scheme have been formerly outlined by Kraft [14] and Hargraves and Paris [11]. A discretization of the time interval 0 = t1 < t2 < . . . < tN = tf (5) is chosen. The parameters Y of the nonlinear program are the values of control and state variables at the grid points tj, j = 1, . . . , N, and the final time tf Y = (u(t1), . . . , u(tN), x(t1), . . . , x(tN ), tN) ∈ IR. (6) The controls are chosen as piecewise linear interpolating functions between u(tj) and u(tj+1) for tj ≤ t < tj+1 uapp(t) = u(tj) + t− tj tj+1 − tj (u(tj+1)− u(tj)). (7) The states are chosen as continuously differentiable functions and piecewise defined as cubic polynomials between x(tj) and x(tj+1) with ẋapp(s) := f(x(s), u(s), s) at s = tj, tj+1,