Approximate optimal control design for quadrotors: A computationally fast solution

Abstract

An approximate closed‐form optimal control design is proposed for the flight control of quadrotor unmanned aerial vehicles. The nonlinear dynamic equation is rewritten as a pseudo‐linear form without approximations. The quadratic cost function is modified by adding perturbation terms to the state weighting matrix. A co‐state, which is associated with the solution to the partial differential Hamilton–Jacobi–Bellman (HJB) equation, is approximated by a power series of an instrumental variable with symmetric matrices as the coefficients. The solution to the intractable HJB equation can be reduced to solving these coefficient matrices, which are in forms of a differential Riccati equation and a series of linear Lyapunov equations, These equations can be solved recursively and analytically. Specifically, the differential Riccati equation can be solved offline and only once, and the linear Lyapunov equations can be solved analytically. These approximations lead to a closed‐form suboptimal state feedback control law, which is computationally more efficient than the similar finite‐time state‐dependent Riccati equation (SDRE) technique that requires the solution of the state‐dependent differential Riccati equation at each time step and demands a high computational cost. The proposed control law is applied to the flight control design of quadrotors. Numerical simulations validate the effectiveness of the proposed optimal control technique with superior performance of control accuracy and robustness. It is compared favorably with the finite‐time SDRE technique in terms of computation efficiency and control effort, especially when onboard implementations and experiments are needed.