An Adaptive Cross Approximation Method for the Hamilton-Jacobi-Bellman Equation * *This work is supported by the National Natural Science Foundation of China (No. 61473233)

Abstract

Abstract The Hamilton-Jacobi-Bellman (HJB) equation can provide closed-loop optimal control feedback, but numerical methods for the HJB equation usually suffer from heavy computational burden. In this paper, an adaptive cross approximation (ACA) based method is proposed to alleviate the heavy computational cost. Firstly, when the HJB equation is discretized and solved on uniform grids, it is shown that under certain assumptions the value function tensor can be obtained elementwise by solving the corresponding two-point boundary value problem (TPBVP). Secondly, the value function tensor of the HJB equation is reshaped into a matrix to incorporate the adaptive cross approximation method. Given an error tolerance, the adaptive cross approximation method can reconstruct the value function tensor with only a small subset of the entries. In the simulation for the four-dimensional spacecraft attitude stabilization problem, compared with calculating the full value function tensor, the proposed method achieves about three orders of magnitude improvement in both computation time and memory requirement. Real-time closed-loop receding horizon control is implemented based on the obtained value function, and it is also compared with the open-loop trajectory. Numerical results show the effectiveness and efficiency of the proposed method.