Differential flatness-based pseudospectral optimal control of six-degrees-of-freedom aircraft and its issues

The advent of efficient numerical algorithms and powerful computing resources have made real-time optimization a reality. However, for systems like 6DoF aircraft, the problem remains challenging due to the complexity and fast dynamics of the system. Smaller optimization problems with fewer constraints can be obtained from a differential flatness-based optimization scheme. This paper proposes a flatness-based nonlinear model predictive controller (NMPC) for a 6DoF aircraft to improve computational time. However, it is difficult to tell in advance if flatness-based NMPC can outperform the simultaneous NMPC with many variables and constraints. This is because there is a trade-off between loss in convexity and an increase in nonlinearity with a dimension reduction. In addition, nonlinear optimization solvers like IPOPT can exploit the underlying sparse structure of the optimization problems from simultaneous NMPC to compute solutions efficiently. Hence, a comparative study between flatness-based and simultaneous nonlinear model predictive control is necessary to assess the computational performance. Results are presented with discussions on solve time, numerical conditioning, convergence, convexity, and solve success rates of the optimization problem. The discussions presented can be extended to other systems to study the effectiveness of flatness-based optimization in a systematic manner. In addition, pseudospectral knots are explored in the paper, which improves sparsity and numerical conditioning of flatness-based optimal control problems.