# Low-Rank Modifications of Riccati Factorizations for Model Predictive Control

## TL;DR

This paper introduces a method to efficiently update Riccati factorizations using low-rank modifications during active-set iterations in MPC, significantly reducing computation time for solving optimal control problems.

## Contribution

The paper develops a structured approach to exploit low-rank changes in Riccati factorizations, improving the efficiency of active-set methods in MPC.

## Key findings

- Significant reduction in computation time demonstrated
- Effective for linear, nonlinear, and hybrid MPC
- Structured low-rank updates outperform full recomputations

## Abstract

In Model Predictive Control (MPC) the control input is computed by solving a constrained finite-time optimal control (CFTOC) problem at each sample in the control loop. The main computational effort is often spent on computing the search directions, which in MPC corresponds to solving unconstrained finite-time optimal control (UFTOC) problems. This is commonly performed using Riccati recursions or generic sparsity exploiting algorithms. In this work the focus is efficient search direction computations for active-set (AS) type methods. The system of equations to be solved at each AS iteration is changed only by a low-rank modification of the previous one, and exploiting this structured change is important for the performance of AS type solvers. In this paper, theory for how to exploit these low-rank changes by modifying the Riccati factorization between AS iterations in a structured way is presented. A numerical evaluation of the proposed algorithm shows that the computation time can be significantly reduced by modifying, instead of re-computing, the Riccati factorization. This speed-up can be important for AS type solvers used for linear, nonlinear and hybrid MPC.

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Source: https://tomesphere.com/paper/1703.07589