Fuzzy Optimal Control Approach in Low-Thrust Orbit Transfer Problem

In this paper, the optimal low thrust planar orbit transfer problem is solved utilizing a fuzzy optimal control algorithm. Firstly, dynamic equations are presented in a discretized form, then all the design variables and constraints are transformed to fuzzy space, while minimizing the performance index and also satisfying transversallity conditions. Applying the concept of membership functions based on expert experience, the designed cost function associated with operational constraints are transformed to fuzzy relations through specific membership functions. Applying Bellman-Zadeh approach, the optimal control problem can be converted to a parameter optimization. Combining the performance index and problem's constraints in a scalar function, necessary optimality conditions are achieved in a form of nonlinear algebraic equations. Finally, to solve this set of equations, the gradient-based method is used. In comparison with the exact form of the problem, the efficiency of the proposed algorithm is highlighted in terms of time and accuracy. In the fuzzy optimal control, a control designer could take advantage of determining the allowed limit for cost function. This algorithm could be successfully extended to fixed state or fixed control problems which is time-consuming in scope of the classical optimal control.