UC Santa Barbara

This work provides a comprehensive review of techniques for solving the discrete algebraic Riccati equation and guidance for selecting an appropriate method for various user-specified DAREs. Motivated by process control, chemical engineering practitioners are driven by safety, profit optimization, and reduced process variability. The solution to the DARE offers a way to regulate and accurately track process behavior through the linear quadratic regulator and the linear quadratic estimator. This study offers a comparison between the iterative Riccati equation (IRE), generalized real-Schur vector methods (GSV), and Newton's method as algorithms for computing the solution to the DARE. Through numerical studies, we find Newton's method struggles as a stand-alone method as it requires information on the solution a priori. For large system sizes (more than 1000 states), the IRE seems to suffer from numerical instability and the GSV severely scales with system size. We offer two switch method alternatives by utilizing Newton's method as a refinement technique on IRE iterations. The A+BK stability switch (ABKSS) prioritizes Newton's method's reliability close to the solution and switches once a stable closed-loops state transfer matrix is found via the IRE. We also pose the IRE proximity switch (IREPS), which computes the iterations to the discrete Riccati equation until iterates are sufficiently close to the stabilizing solution and refined through iterations of Newton's method. For randomly generated systems of 1000 or more states, both switch methods offer a high-accuracy alternative that surpasses the GSV in computation time and circumvents the numerical instability experienced by the IRE. Furthermore, we find that the GSV and IRE are highly susceptible to systems that are close to being unstabilizable (nearly unstable directions of state matrix or nearly zero influence in the control matrix). IREPS and ABKSS offer significant improvements for computing the solution to barely stabilizable systems. Lastly, we initiate an investigation into utilizing Newton's method for DAREs with singular R. While not as general as positive semidefinite R, we prove that Newton's method can compute the solution to DAREs with R equal to zero as a special case. The purpose of this study is to revisit methods for the DARE with modernized computation tools. We find that as we pose more challenging problems, standard algorithms for computing the solution to the DARE become unviable and require a new approach with these old tools.