Flatness-Based Reduced Hessian Method for Optimal Control of Aircraft

Numerical solutions of flatness-based reformulation of optimal control problems lead to an optimization problem with fewer variables and constraints. However, the expressions for the states and controls in terms of the flat output variables can be highly nonlinear and complicated. Hence, this may lead to expensive function evaluations within the optimization problem, degradation of convexity, and issues with convergence. Thus, a flatness-based reformulation of the optimal control problems may not be a viable alternative for computational guidance and control. Alternatively, a novel flatness-based reduced Hessian sequential quadratic programming algorithm is developed in this paper to solve optimization problems for a six-degree-of-freedom aircraft. An analytical null space basis is derived from the linearized differential constraints, which leads to a reduced-dimensional quadratic program with the discretized flat outputs as decision variables. Unlike flatness-based reformulation of the optimal control problem, the null space/reduced Hessian method does not introduce additional nonlinearity and preserves the convexity of the optimization problem. A nonlinear model predictive control problem is solved to demonstrate the reduced Hessian algorithm. The algorithm is validated and Monte Carlo trials are performed to assess the effectiveness of the approach. Computational studies show that the current approach is faster than methods that directly exploit sparsity up to a factor of five.