DIRECT OPTIMIZATION USING COLLOCATION BASED ON HIGH-ORDER GAUSS-LOBATTO QUADRATURE-RULES

The method of collocation and nonlinear programming has been used rece ntly to solve a number of optimal control problems. In this method pol ynomials are commonly used to represent the state variable time histor ies over subintervals of the total time of interest. These polynomials correspond to a family of modified-Gaussian quadrature rules known as the Gauss-Lobatto rules. Presently, relatively low-order rules from t he Gauss-Lobatto family, such as the trapezoid and Simpson's rule, are used to construct collocation solution schemes. In this work higher-o rder Gauss-Lobatto quadrature rules are formulated using collocation p oint selection based on a particular family of Jacobi polynomials. The advantage of using a quadrature rule of higher order is that the appr oximation using the higher degree polynomial may be more accurate, due to finite precision arithmetic, than a formulation based on a lower d egree polynomial. In addition, the number of subintervals and, therefo re, the number of nonlinear programming parameters needed to solve a p roblem accurately may he significantly reduced from that required if t he conventional trapezoidal or Simpson's quadrature schemes are used, An optimal trajectory maximizing final energy for a low-thrust spacecr aft is used to demonstrate the benefits of using the higher-order sche mes.