Inverse Linear Quadratic Regulator Problem for Discrete-Time Systems via LMI

This paper focuses on the inverse optimal control problem for discrete-time systems with unknown cost functions using linear matrix inequalities (LMIs). Based on Pontryagin's minimum principle, a novel model-based algorithm is proposed to reconstruct the weight matrices of an unknown quadratic cost function. This algorithm is formulated using LMIs to find a solution to the algebraic Riccati equation, which is then used to determine the weight matrices. We prove that the solution obtained from LMIs belongs to the set of solutions for the inverse optimal control problem. On this basis, a noniterative model-free algorithm is presented to solve the problem without requiring system matrices. The collected data is used to estimate the unknown weight matrices, and we prove that our algorithm can obtain an approximate optimal solution. Finally, an example illustrates the effectiveness of our algorithms, and comparisons show that our algorithms outperform others.