The Inverse Problem of Continuous-Time Linear Quadratic Gaussian Control With Application to Biological Systems Analysis

In this paper, we demonstrate two methods for solving the inverse problem of continuous-time LQG control. This problem can be defined as: given a known LTI system with feedback controller K and Kalman gain L, can we find the weighting matrices Q, R (for state and input, respectively) and estimated noise intensities W, V (for process and measurement noise, respectively) such that the LQG control synthesis problem using these weights generates K and L? We formulate a regularized version of this problem as a minimization problem subject to a set of Linear Matrix Inequalities (LMIs). If feasible, a unique exact solution to the inverse LQR problem exists. If the LMIs are infeasible, we show a gradient descent algorithm that will find Q, R, W, and V to minimize the error in the recovered gain matrices K and L. We demonstrate these techniques through several numerical examples and formulate a human postural control case study to which we intend to apply our proposed techniques.