Solutions to the Inverse LQR Problem with Application to Biological Systems Analysis

Area of Science:

• Control Systems Engineering
• Computational Neuroscience
• Robotics

Background:

• The Linear Quadratic Regulator (LQR) is a fundamental optimal control problem.
• Understanding the underlying cost function is crucial for analyzing control systems, particularly in biological contexts.
• The inverse LQR problem seeks to identify system weights from a given controller.

Purpose of the Study:

• To develop techniques for determining the cost function of time-invariant LQR problems.
• To address the inverse LQR problem: identifying weighting matrices Q and R for a given controller K.
• To analyze motion goals in biological systems by recovering their cost functions.

Main Methods:

• Utilized Linear Matrix Inequality (LMI) methods for solving the inverse LQR problem with unknown Q and R.
• Proposed a gradient-based, least-squares minimization method for cases where LMIs are infeasible due to estimation errors.
• Developed an LMI minimization problem to find optimal initial points for the gradient descent algorithm.

Main Results:

• An efficient LMI method was presented, providing a unique solution when feasible.
• The gradient-based method effectively approximates solutions for infeasible LMI cases, handling noisy experimental data.
• Successfully applied the technique to human posture control on a moving robot, recovering a relevant cost function.

Conclusions:

• The developed techniques provide robust methods for solving the inverse LQR problem in continuous and discrete time.
• The approach offers valuable insights into human motor control by enabling cost function recovery.
• This work has significant implications for control theory, neuroscience, and robotics applications.