title: Solutions to the Inverse LQR Problem with Application to Biological Systems Analysis.
abstract: In this paper, we present a set of techniques for finding a cost function to the time-invariant Linear Quadratic Regulator (LQR) problem in both continuous- and discrete-time cases. Our methodology is based on the solution to the inverse LQR problem, which can be stated as: does a given controllerKdescribe the solution to a time-invariant LQR problem, and if so, what weightsQandRproduceKas the optimal solution? Our motivation for investigating this problem is the analysis of motion goals in biological systems. We first describe an efficient Linear Matrix Inequality (LMI) method for determining a solution to the general case of this inverse LQR problem when both the weighting matricesQandRare unknown. Our first LMI-based formulation provides a unique solution when it is feasible. Additionally, we propose a gradient-based, least-squares minimization method that can be applied to approximate a solution in cases when the LMIs are infeasible. This new method is very useful in practice since the estimated gain matrixKfrom the noisy experimental data could be perturbed by the estimation error, which may result in the infeasibility of the LMIs. We also provide an LMI minimization problem to find a good initial point for the minimization using the proposed gradient descent algorithm. We then provide a set of examples to illustrate how to apply our approaches to several different types of problems. An important result is the application of the technique to human subject posture control when seated on a moving robot. Results show that we can recover a cost function which may provide a useful insight on the human motor control goal.
authors.affiliation: Michigan State University Dept. of Mechanical Engineering and the MSU Center for Orthopedic Research (MSUCOR), East Lansing, MI 48824.
authors.forename: M Cody
authors.initials: MC
authors.lastname: Priess
authors.affiliation: Western Digital Inc.
authors.forename: Richard
authors.initials: R
authors.lastname: Conway
authors.affiliation: MSU Dept. of Mechanical Engineering, Dept. of Electrical and Computer Engineering, and MSUCOR.
authors.forename: Jongeun
authors.initials: J
authors.lastname: Choi
authors.affiliation: MSU Dept. of Surgical Specialties and MSUCOR.
authors.forename: John M
authors.initials: JM
authors.lastname: Popovich
authors.affiliation: Michigan State University Dept. of Mechanical Engineering and the MSU Center for Orthopedic Research (MSUCOR), East Lansing, MI 48824.
authors.forename: Clark
authors.initials: C
authors.lastname: Radcliffe