Data-driven Estimation of Algebraic Riccati Equation for Inverse Linear Quadratic Regulator Problem

In this research, we propose a method to estimate the algebraic Riccati equation (ARE) under the assumption that the system is unknown, but its state and input can be observed. The ARE is essential to solve the inverse linear quadratic regulator (ILQR) problem. The ILQR problem is a problem of finding the quadratic objective function optimized by the given linear time-invariant feedback system. The ILQR problem is easily solved by solving the linear matrix inequality including the ARE if the system model is readily available. However, the system is often unknown in biological system analysis, the main fields where the ILQR problem is expected to be applied. To solve this problem, we obtain an equation that is derived from the ARE but only requires trajectory data of the system and not the system model. We further provide two conditions for the equation to be equivalent to the original ARE. The advantage of our method compared to system identification is that prior information on the objective function can be used to reduce the data needed for ARE estimation. Numerical experiments are performed to support our claims.