Optimization of Weight Matrices for the Linear Quadratic Regulator Problem Using Algebraic Closed-Form Solutions

This work proposes an analytical gradient-based optimization approach to determine the optimal weight matrices that make the state and control input at the final time close to zero for the linear quadratic regulator problem. Most existing methodologies focused on regulating the diagonal elements using only bio-inspired approaches or analytical approaches. The method proposed, contrarily, optimizes both diagonal and off-diagonal matrix elements based on the gradient. Moreover, by introducing a new variable composed of the steady-state and time-varying terms for the Riccati matrix and using the coordinate transformation for the state, one develops algebraic equationsbased closed-form solutions to generate the required states and numerical partial derivatives for an optimization strategy that does not require the computationally intensive numerical integration process. The authors test the algorithm with one- and two-degrees-of-freedom linear plant models, and it yields the weight matrices that successfully satisfy the pre-defined requirement, which is the norm of the augmented states less than 10⁻⁵. The results suggest the broad applicability of the proposed algorithm in science and engineering.