Brunovsky Riccati Recursion for Linear Model Predictive Control

In almost all algorithms for Model Predictive Control (MPC), the most time-consuming step is to solve some form of Linear Quadratic (LQ) Optimal Control Problem (OCP) repeatedly. The commonly recognized best option for this is a Riccati recursion based solver, which has a time complexity of $\mathcal{O}\left( {N\left( {n_x^3 + n_x^2{n_u} + {n_x}n_u^2 + n_u^3} \right)} \right)$. In this paper, we propose a novel Brunovsky Riccati Recursion algorithm to solve LQ OCPs for Linear Time Invariant (LTI) systems. The algorithm transforms the system into Brunovsky form, formulates a new LQ cost (and constraints, if any) in Brunovsky coordinates, performs the Riccati recursion there, and converts the solution back. Due to the sparsity (block-diagonality and zero-one pattern per block) of Brunovsky form and the data parallelism introduced in the cost, constraints, and solution transformations, the time complexity of the new method is greatly reduced to $\mathcal{O}\left( {n_x^3 + N\left( {n_x^2{n_u} + {n_x}n_u^2 + n_u^3} \right)} \right)$ if N threads/cores are available for parallel computing.